GPT-4o on 4.MD.A.3
Common Core Standard 4.MD.A.3
This standard covers the following:
Apply the area and perimeter formulas for rectangles in real world and mathematical problems.
Problem: A rectangle has a length of 21m and a width of 37m. What is the perimeter of the rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 21 meters
- Width (\(W\)) = 37 meters
First, we add the length and the width:
\[ L + W = 21 + 37 = 58 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ \text{Perimeter} = 2 \times 58 = 116 \]
### Answer: 116 meters
Expected answer: 116
Correct? Yes
Problem: Now consider that the length of the rectangle has changed to 4m while the width remains at 37m. What is the new perimeter of the rectangle?
GPT-4o:
To find the new perimeter of the rectangle with the updated dimensions, we use the same perimeter formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- New Length (\(L\)) = 4 meters
- Width (\(W\)) = 37 meters
First, we add the new length and the width:
\[ L + W = 4 + 37 = 41 \]
Next, we multiply the sum by 2 to find the new perimeter:
\[ \text{Perimeter} = 2 \times 41 = 82 \]
### Answer: 82 meters
Expected answer: 82
Correct? Yes
Problem: A rectangular garden has a length of 50 feet and a width of 20 feet. What is the perimeter of the garden?
GPT-4o:
To find the perimeter of a rectangular garden, we use the formula for the perimeter of a rectangle, which is:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length = 50 feet
- Width = 20 feet
First, we add the length and the width:
\[ 50 + 20 = 70 \]
Next, we multiply the sum by 2:
\[ 2 \times 70 = 140 \]
Therefore, the perimeter of the garden is 140 feet.
### Answer: 140 feet
Expected answer: 140
Correct? Yes
Problem: A landscaper plans to reduce the length of the rectangular garden to 15 feet while keeping the width the same. What will the new perimeter of the garden be?
GPT-4o:
To find the new perimeter of the rectangular garden after reducing the length to 15 feet while keeping the width the same, we use the same perimeter formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- New Length = 15 feet
- Width = 20 feet
First, we add the new length and the width:
\[ 15 + 20 = 35 \]
Next, we multiply the sum by 2:
\[ 2 \times 35 = 70 \]
Therefore, the new perimeter of the garden will be 70 feet.
### Answer: 70 feet
Expected answer: 70
Correct? Yes
Problem: A rectangle has a width of 7m and a length of 26m. What is the area of this rectangle?
GPT-4o:
To find the area of a rectangle, we use the formula:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Given:
- Width = 7 meters
- Length = 26 meters
Substitute the given values into the formula:
\[ \text{Area} = 26 \, \text{m} \times 7 \, \text{m} \]
Calculate the product:
\[ \text{Area} = 182 \, \text{m}^2 \]
### Answer: 182 m²
Expected answer: 182
Correct? Yes
Problem: The width of the rectangle stays the same at 7m, but the length extends by 9m to become 35m. What is the area of the rectangle now?
GPT-4o:
To find the new area of the rectangle, we use the same formula for the area of a rectangle:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Given:
- Width = 7 meters
- New Length = 35 meters (since the original length of 26 meters is extended by 9 meters)
Substitute the given values into the formula:
\[ \text{Area} = 35 \, \text{m} \times 7 \, \text{m} \]
Calculate the product:
\[ \text{Area} = 245 \, \text{m}^2 \]
### Answer: 245 m²
Expected answer: 245
Correct? Yes
Problem: A rectangle has a length of 50m and a width of 15m. What is its perimeter?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 50 meters
- Width (\(W\)) = 15 meters
First, we add the length and the width:
\[ L + W = 50 + 15 = 65 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ \text{Perimeter} = 2 \times 65 = 130 \]
### Answer: 130 meters
Expected answer: 130
Correct? Yes
Problem: The rectangle mentioned in the previous problem had its width reduced to 9m while the length remained the same. What is the new perimeter of the rectangle?
GPT-4o:
To find the new perimeter of the rectangle after the width is reduced, we use the same perimeter formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 50 meters (unchanged)
- New Width (\(W\)) = 9 meters
First, we add the length and the new width:
\[ L + W = 50 + 9 = 59 \]
Next, we multiply the sum by 2 to find the new perimeter:
\[ \text{Perimeter} = 2 \times 59 = 118 \]
### Answer: 118 meters
Expected answer: 118
Correct? Yes
Problem: A rectangular fence has a side length of 5m and another side of 19m. What is the total distance, in meters, around the fence (perimeter)?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{length} + \text{width}) \]
Given:
- Length = 19 meters
- Width = 5 meters
First, we add the length and the width:
\[ 19 + 5 = 24 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ 2 \times 24 = 48 \]
### Answer: 48 meters
Expected answer: 48
Correct? Yes
Problem: The owner of the fence decided to extend the side that originally measured 19m to a new length of 36m, while the other side remains the same. After the extension, what will be the new total distance, in meters, around the fence (perimeter)?
GPT-4o:
To find the new perimeter of the rectangle after the extension, we use the same perimeter formula:
\[ \text{Perimeter} = 2 \times (\text{length} + \text{width}) \]
Given:
- New length = 36 meters
- Width = 5 meters
First, we add the new length and the width:
\[ 36 + 5 = 41 \]
Next, we multiply the sum by 2 to find the new perimeter:
\[ 2 \times 41 = 82 \]
### Answer: 82 meters
Expected answer: 82
Correct? Yes
Problem: A rectangle has sides of length 49m and 6m. What is the perimeter of the rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{length} + \text{width}) \]
Given the length \( l = 49 \) meters and the width \( w = 6 \) meters, we can substitute these values into the formula:
\[ \text{Perimeter} = 2 \times (49 + 6) \]
First, calculate the sum inside the parentheses:
\[ 49 + 6 = 55 \]
Next, multiply by 2:
\[ 2 \times 55 = 110 \]
### Answer: 110 meters
Expected answer: 110
Correct? Yes
Problem: Suppose we were mistaken in the original measurement, and one side of the rectangle was actually 17m instead of 49m. What would the new perimeter of the rectangle be?
GPT-4o:
To find the new perimeter of the rectangle with the corrected side lengths, we use the same formula for the perimeter of a rectangle:
\[ \text{Perimeter} = 2 \times (\text{length} + \text{width}) \]
Given the corrected length \( l = 17 \) meters and the width \( w = 6 \) meters, we substitute these values into the formula:
\[ \text{Perimeter} = 2 \times (17 + 6) \]
First, calculate the sum inside the parentheses:
\[ 17 + 6 = 23 \]
Next, multiply by 2:
\[ 2 \times 23 = 46 \]
### Answer: 46 meters
Expected answer: 46
Correct? Yes
Problem: Joshua plans to lay a wire around his rectangular garden. If the garden is 7m wide and 26m long, how much wire will Joshua need to enclose the garden completely?
GPT-4o:
To determine the amount of wire Joshua will need to enclose his rectangular garden, we need to calculate the perimeter of the garden. The formula for the perimeter \( P \) of a rectangle is given by:
\[ P = 2 \times (\text{length} + \text{width}) \]
Given:
- Length (\( l \)) = 26 meters
- Width (\( w \)) = 7 meters
Substitute the given values into the formula:
\[ P = 2 \times (26 + 7) \]
\[ P = 2 \times 33 \]
\[ P = 66 \]
Therefore, Joshua will need 66 meters of wire to enclose his garden completely.
### Answer: 66 meters
Expected answer: 66
Correct? Yes
Problem: A rectangle has a length of 18 meters and a width of 11 meters. What is the perimeter of the rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length = 18 meters
- Width = 11 meters
First, we add the length and the width:
\[ 18 + 11 = 29 \]
Next, we multiply the sum by 2:
\[ 2 \times 29 = 58 \]
### Answer: 58 meters
Expected answer: 58
Correct? Yes
Problem: Suppose the width of the rectangle changed to 44 meters, while the length remained at 18 meters. What would be the new perimeter of the rectangle?
GPT-4o:
To find the new perimeter of the rectangle with the updated width, we use the same perimeter formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length = 18 meters
- Width = 44 meters
First, we add the length and the width:
\[ 18 + 44 = 62 \]
Next, we multiply the sum by 2:
\[ 2 \times 62 = 124 \]
### Answer: 124 meters
Expected answer: 124
Correct? Yes
Problem: The dimensions of a rectangle are 22 feet by 39 feet. What is the perimeter of the rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given the dimensions of the rectangle:
- Length = 39 feet
- Width = 22 feet
First, we add the length and the width:
\[ 39 + 22 = 61 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ 2 \times 61 = 122 \]
### Answer: 122 feet
Expected answer: 122
Correct? Yes
Problem: If the length of the rectangle is now 26 feet instead of 39 feet, while the width remains the same at 22 feet, what would then be the new perimeter of the rectangle?
GPT-4o:
To find the new perimeter of the rectangle with the updated dimensions, we use the same formula for the perimeter of a rectangle:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given the new dimensions:
- Length = 26 feet
- Width = 22 feet
First, we add the length and the width:
\[ 26 + 22 = 48 \]
Next, we multiply the sum by 2 to find the new perimeter:
\[ 2 \times 48 = 96 \]
### Answer: 96 feet
Expected answer: 96
Correct? Yes
Problem: A rectangle has two sides measuring 45m and the other two sides measuring 25m each. What is the perimeter of this rectangle?
GPT-4o:
To find the perimeter of a rectangle, we add up the lengths of all four sides.
Given:
- Two sides measure 45 meters each.
- The other two sides measure 25 meters each.
The formula for the perimeter \( P \) of a rectangle is:
\[ P = 2 \times (\text{length} + \text{width}) \]
Substituting the given values:
\[ P = 2 \times (45 \, \text{m} + 25 \, \text{m}) \]
\[ P = 2 \times 70 \, \text{m} \]
\[ P = 140 \, \text{m} \]
### Answer: 140 meters.
Expected answer: 140
Correct? Yes
Problem: In the previous rectangle question, suppose one of the sides measuring 45m was incorrectly assessed. The actual measurement is found to be 42m. With this new information, what would be the adjusted perimeter of the rectangle?
GPT-4o:
To find the adjusted perimeter of the rectangle with the corrected side measurement, we need to use the new side lengths.
Given:
- One side is corrected to 42 meters.
- The other side remains 45 meters.
- The other two sides measure 25 meters each.
The formula for the perimeter \( P \) of a rectangle is:
\[ P = 2 \times (\text{length} + \text{width}) \]
Substituting the corrected values:
\[ P = 2 \times (42 \, \text{m} + 25 \, \text{m}) \]
\[ P = 2 \times 67 \, \text{m} \]
\[ P = 134 \, \text{m} \]
### Answer: 134 meters.
Expected answer: 134
Correct? Yes
Problem: A rectangular garden has a length of 8m and a width of 5m. What is the perimeter of this garden?
GPT-4o:
To find the perimeter of a rectangular garden, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 8 meters
- Width (\(W\)) = 5 meters
Substitute the given values into the formula:
\[ \text{Perimeter} = 2 \times (8 + 5) \]
\[ \text{Perimeter} = 2 \times 13 \]
\[ \text{Perimeter} = 26 \]
### Answer: 26 meters
Expected answer: 26
Correct? Yes
Problem: The owner of the rectangular garden decided to extend its length to 46m while keeping the width the same, i.e., 5m. What is the new perimeter of the garden now?
GPT-4o:
To find the new perimeter of the rectangular garden after extending its length, we use the same perimeter formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- New Length (\(L\)) = 46 meters
- Width (\(W\)) = 5 meters
Substitute the new length and the width into the formula:
\[ \text{Perimeter} = 2 \times (46 + 5) \]
\[ \text{Perimeter} = 2 \times 51 \]
\[ \text{Perimeter} = 102 \]
### Answer: 102 meters
Expected answer: 102
Correct? Yes
Problem: In a rectangle, the length is 19 units and the width is 17 units. What is the area of this rectangle?
GPT-4o:
To find the area of a rectangle, we use the formula:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Given:
- Length = 19 units
- Width = 17 units
Substitute the given values into the formula:
\[ \text{Area} = 19 \times 17 \]
Now, perform the multiplication:
\[ 19 \times 17 = 323 \]
### Answer: 323 units²
Expected answer: 323
Correct? Yes
Problem: In the previous rectangle problem, what if the length was 22 units instead of 19 units while the width remained the same at 17 units? What would be the new area of the rectangle?
GPT-4o:
To find the new area of the rectangle with the updated length, we use the same area formula:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Given:
- New Length = 22 units
- Width = 17 units
Substitute the new length and the given width into the formula:
\[ \text{Area} = 22 \times 17 \]
Now, perform the multiplication:
\[ 22 \times 17 = 374 \]
### Answer: 374 units²
Expected answer: 374
Correct? Yes
Problem: A rectangle has a length of 42 meters and a width of 8 meters. What is the perimeter of the rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\( L \)) = 42 meters
- Width (\( W \)) = 8 meters
First, we add the length and the width:
\[ L + W = 42 + 8 = 50 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ \text{Perimeter} = 2 \times 50 = 100 \]
### Answer: 100 meters
Expected answer: 100
Correct? Yes
Problem: A rectangle has a width of 16m and a length of 38m. What is the area of this rectangle?
GPT-4o:
To find the area of a rectangle, we use the formula:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Given:
- Width = 16 meters
- Length = 38 meters
Substitute the given values into the formula:
\[ \text{Area} = 38 \, \text{m} \times 16 \, \text{m} \]
Now, perform the multiplication:
\[ 38 \times 16 = 608 \]
So, the area of the rectangle is 608 square meters.
### Answer: 608 square meters.
Expected answer: 608
Correct? Yes
Problem: In the rectangle from the previous problem, suppose we replace the length of 38m with a new length of 22m. What would the new area of the rectangle be?
GPT-4o:
To find the new area of the rectangle with the updated length, we use the same area formula:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Given:
- New Length = 22 meters
- Width = 16 meters
Substitute the new length and the width into the formula:
\[ \text{Area} = 22 \, \text{m} \times 16 \, \text{m} \]
Now, perform the multiplication:
\[ 22 \times 16 = 352 \]
So, the new area of the rectangle is 352 square meters.
### Answer: 352 square meters.
Expected answer: 352
Correct? Yes
Problem: Consider a rectangular plot of land. The plot has a length of 28m and a width of 17m. What is the area of the plot?
GPT-4o:
To find the area of a rectangular plot of land, we use the formula for the area of a rectangle, which is:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Given:
- Length = 28 meters
- Width = 17 meters
Substituting the given values into the formula:
\[ \text{Area} = 28 \, \text{m} \times 17 \, \text{m} \]
Now, we perform the multiplication:
\[ 28 \times 17 = 476 \]
Thus, the area of the rectangular plot of land is 476 square meters.
### Answer: 476 square meters
Expected answer: 476
Correct? Yes
Problem: Suppose instead of the original 28m, the length of the rectangular plot of land were only 10m while keeping the width the same at 17m. What would then be the area of the rectangular plot of land?
GPT-4o:
To find the area of the rectangular plot of land with the new dimensions, we again use the formula for the area of a rectangle:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Given the new dimensions:
- Length = 10 meters
- Width = 17 meters
Substituting the given values into the formula:
\[ \text{Area} = 10 \, \text{m} \times 17 \, \text{m} \]
Now, we perform the multiplication:
\[ 10 \times 17 = 170 \]
Thus, the area of the rectangular plot of land with the new dimensions is 170 square meters.
### Answer: 170 square meters
Expected answer: 170
Correct? Yes
Problem: A rectangle has a length of 43m and a width of 41m. What is the perimeter of this rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 43 meters
- Width (\(W\)) = 41 meters
First, we add the length and the width:
\[ L + W = 43 + 41 = 84 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ \text{Perimeter} = 2 \times 84 = 168 \]
### Answer: 168 meters
Expected answer: 168
Correct? Yes
Problem: Suppose the rectangle from the previous problem had its length reduced to 18m while keeping the width the same. What would be the new perimeter of the rectangle?
GPT-4o:
To find the new perimeter of the rectangle with the updated dimensions, we use the same perimeter formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given the new dimensions:
- New Length (\(L\)) = 18 meters
- Width (\(W\)) = 41 meters
First, we add the new length and the width:
\[ L + W = 18 + 41 = 59 \]
Next, we multiply the sum by 2 to find the new perimeter:
\[ \text{Perimeter} = 2 \times 59 = 118 \]
### Answer: 118 meters
Expected answer: 118
Correct? Yes
Problem: A rectangle has one side that measures 44m, and the other side that measures 17m. What is the perimeter of the rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{length} + \text{width}) \]
Given:
- Length = 44 meters
- Width = 17 meters
First, we add the length and the width:
\[ 44 + 17 = 61 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ 2 \times 61 = 122 \]
### Answer: 122 meters
Expected answer: 122
Correct? Yes
Problem: In the previous problem, we considered a rectangle with one side that measures 44m, and the other side that measures 17m. What would be the perimeter of the rectangle if instead the length of the second side were 21m?
GPT-4o:
To find the perimeter of the rectangle with the new dimensions, we use the same formula:
\[ \text{Perimeter} = 2 \times (\text{length} + \text{width}) \]
Given:
- Length = 44 meters
- New Width = 21 meters
First, we add the length and the new width:
\[ 44 + 21 = 65 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ 2 \times 65 = 130 \]
### Answer: 130 meters
Expected answer: 130
Correct? Yes
Problem: A rectangle's perimeter is twice the sum of its width and length. If the rectangle's width is 2m and its length is 20m, what is its perimeter?
GPT-4o:
To find the perimeter of the rectangle, we use the formula for the perimeter of a rectangle, which is given by:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Width (\( W \)) = 2 meters
- Length (\( L \)) = 20 meters
First, we calculate the sum of the length and the width:
\[ L + W = 20 + 2 = 22 \text{ meters} \]
Next, we multiply this sum by 2 to find the perimeter:
\[ \text{Perimeter} = 2 \times 22 = 44 \text{ meters} \]
### Answer: 44 meters
Expected answer: 44
Correct? Yes
Problem: If the width of the rectangle is increased from 2m to 32m, while keeping the length the same at 20m, what is the new perimeter of the rectangle?
GPT-4o:
To find the new perimeter of the rectangle after increasing the width, we use the same perimeter formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- New Width (\( W \)) = 32 meters
- Length (\( L \)) = 20 meters
First, we calculate the sum of the new width and the length:
\[ L + W = 20 + 32 = 52 \text{ meters} \]
Next, we multiply this sum by 2 to find the new perimeter:
\[ \text{Perimeter} = 2 \times 52 = 104 \text{ meters} \]
### Answer: 104 meters
Expected answer: 104
Correct? Yes
Problem: A rectangle has a length of 4 meters and a width of 5 meters. What is the perimeter of the rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length = 4 meters
- Width = 5 meters
First, we add the length and the width:
\[ 4 + 5 = 9 \]
Next, we multiply the sum by 2:
\[ 2 \times 9 = 18 \]
So, the perimeter of the rectangle is 18 meters.
### Answer: 18 meters
Expected answer: 18
Correct? Yes
Problem: Consider the same rectangle from before, but now suppose its length is extended to 42 meters while the width remains 5 meters. What would then be the perimeter of the rectangle?
GPT-4o:
To find the perimeter of the rectangle with the new dimensions, we use the same formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- New Length = 42 meters
- Width = 5 meters
First, we add the new length and the width:
\[ 42 + 5 = 47 \]
Next, we multiply the sum by 2:
\[ 2 \times 47 = 94 \]
So, the perimeter of the rectangle with the new dimensions is 94 meters.
### Answer: 94 meters
Expected answer: 94
Correct? Yes
Problem: A rectangle has a length of 45m and a width of 37m. What is the perimeter of this rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 45 meters
- Width (\(W\)) = 37 meters
First, we add the length and the width:
\[ L + W = 45 + 37 = 82 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ \text{Perimeter} = 2 \times 82 = 164 \]
### Answer: 164 meters
Expected answer: 164
Correct? Yes
Problem: If the length of the rectangle remains the same at 45m, but the width is increased to 47m, what would be the new perimeter of the rectangle?
GPT-4o:
To find the new perimeter of the rectangle with the updated width, we use the same perimeter formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 45 meters
- New Width (\(W\)) = 47 meters
First, we add the length and the new width:
\[ L + W = 45 + 47 = 92 \]
Next, we multiply the sum by 2 to find the new perimeter:
\[ \text{Perimeter} = 2 \times 92 = 184 \]
### Answer: 184 meters
Expected answer: 184
Correct? Yes
Problem: A rectangle has a length of 27m and a width of 21m. What is the perimeter of this rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 27 meters
- Width (\(W\)) = 21 meters
First, we add the length and the width:
\[ L + W = 27 + 21 = 48 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ \text{Perimeter} = 2 \times 48 = 96 \]
### Answer: 96 meters
Expected answer: 96
Correct? Yes
Problem: Suppose we extended the length of the rectangle from the previous problem to 45m, while keeping the width at 21m. What would the new perimeter of this rectangle be?
GPT-4o:
To find the new perimeter of the rectangle with the updated dimensions, we use the same perimeter formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given the new dimensions:
- Length (\(L\)) = 45 meters
- Width (\(W\)) = 21 meters
First, we add the new length and the width:
\[ L + W = 45 + 21 = 66 \]
Next, we multiply the sum by 2 to find the new perimeter:
\[ \text{Perimeter} = 2 \times 66 = 132 \]
### Answer: 132 meters
Expected answer: 132
Correct? Yes
Problem: A rectangular garden has a length of 30m and a width of 20m. What is the perimeter of the garden?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 30 meters
- Width (\(W\)) = 20 meters
Substitute the given values into the formula:
\[ \text{Perimeter} = 2 \times (30 + 20) \]
\[ \text{Perimeter} = 2 \times 50 \]
\[ \text{Perimeter} = 100 \]
### Answer: 100 meters
Expected answer: 100
Correct? Yes
Problem: Suppose the width of the rectangular garden is increased to 48m, while the length remains the same at 30m. What would be the new perimeter of the garden?
GPT-4o:
To find the new perimeter of the rectangular garden with the updated width, we use the same perimeter formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 30 meters
- New Width (\(W\)) = 48 meters
Substitute the given values into the formula:
\[ \text{Perimeter} = 2 \times (30 + 48) \]
\[ \text{Perimeter} = 2 \times 78 \]
\[ \text{Perimeter} = 156 \]
### Answer: 156 meters
Expected answer: 156
Correct? Yes
Problem: A rectangle has a length of 30m and a width of 27m. What is the perimeter of this rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 30 meters
- Width (\(W\)) = 27 meters
First, we add the length and the width:
\[ L + W = 30 + 27 = 57 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ \text{Perimeter} = 2 \times 57 = 114 \]
### Answer: 114 meters
Expected answer: 114
Correct? Yes
Problem: A rectangle has a length of 23m and a width of 11m. What is the area of this rectangle?
GPT-4o:
To find the area of a rectangle, we use the formula:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Given:
- Length = 23 meters
- Width = 11 meters
Substitute the given values into the formula:
\[ \text{Area} = 23 \, \text{m} \times 11 \, \text{m} \]
Calculate the product:
\[ \text{Area} = 253 \, \text{m}^2 \]
### Answer: 253 m²
Expected answer: 253
Correct? Yes
Problem: A rectangle has a length of 3m and a width of 9m. What would be the total perimeter of this rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 3 meters
- Width (\(W\)) = 9 meters
First, we add the length and the width:
\[ L + W = 3 + 9 = 12 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ \text{Perimeter} = 2 \times 12 = 24 \]
### Answer: 24 meters
Expected answer: 24
Correct? Yes
Problem: Continuing from the previous problem, suppose the length of the rectangle is still 3m, but now the width has increased to 13m. What is the new total perimeter of the rectangle?
GPT-4o:
To find the new perimeter of the rectangle with the updated width, we use the same formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 3 meters
- New Width (\(W\)) = 13 meters
First, we add the length and the new width:
\[ L + W = 3 + 13 = 16 \]
Next, we multiply the sum by 2 to find the new perimeter:
\[ \text{Perimeter} = 2 \times 16 = 32 \]
### Answer: 32 meters
Expected answer: 32
Correct? Yes
Problem: A rectangle has a length of 31m and a width of 6m. What is the perimeter of this rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 31 meters
- Width (\(W\)) = 6 meters
First, we add the length and the width:
\[ L + W = 31 + 6 = 37 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ \text{Perimeter} = 2 \times 37 = 74 \]
### Answer: 74 meters
Expected answer: 74
Correct? Yes
Problem: Now consider, if instead of a length of 31m, the rectangle had a length of 43m but the width remained the same at 6m. What would be the new perimeter of this rectangle?
GPT-4o:
To find the new perimeter of the rectangle with the updated length, we use the same perimeter formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- New Length (\(L\)) = 43 meters
- Width (\(W\)) = 6 meters
First, we add the new length and the width:
\[ L + W = 43 + 6 = 49 \]
Next, we multiply the sum by 2 to find the new perimeter:
\[ \text{Perimeter} = 2 \times 49 = 98 \]
### Answer: 98 meters
Expected answer: 98
Correct? Yes
Problem: A rectangle has a length of 12m and a width of 33m. What is the area of this rectangle?
GPT-4o:
To find the area of a rectangle, we use the formula:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Given:
- Length = 12 meters
- Width = 33 meters
Substitute the given values into the formula:
\[ \text{Area} = 12 \, \text{m} \times 33 \, \text{m} \]
Calculate the product:
\[ \text{Area} = 396 \, \text{m}^2 \]
### Answer: 396 m²
Expected answer: 396
Correct? Yes
Problem: Now, consider that the length of the rectangle was initially 18m instead of 12m, while the width remains the same at 33m. What would be the area of the rectangle in this scenario?
GPT-4o:
To find the area of the rectangle with the new length, we use the same formula:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Given:
- New Length = 18 meters
- Width = 33 meters
Substitute the given values into the formula:
\[ \text{Area} = 18 \, \text{m} \times 33 \, \text{m} \]
Calculate the product:
\[ \text{Area} = 594 \, \text{m}^2 \]
### Answer: 594 m²
Expected answer: 594
Correct? Yes
Problem: A rectangle has a length of 20m and all of its sides are of equal length. What is the perimeter of this rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given that the length of the rectangle is 20 meters, we need to know the width to proceed. However, the problem states that "all of its sides are of equal length," which implies that the shape is actually a square, not a rectangle. Therefore, each side of the square is 20 meters.
For a square, the perimeter is calculated as:
\[ \text{Perimeter} = 4 \times \text{Side length} \]
Substituting the given side length:
\[ \text{Perimeter} = 4 \times 20 \]
\[ \text{Perimeter} = 80 \]
### Answer: 80 meters
Expected answer: 80
Correct? Yes
Problem: A rectangular playground has a length of 35m and a width of 9m. What is the perimeter of the playground?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 35 meters
- Width (\(W\)) = 9 meters
First, we add the length and the width:
\[ L + W = 35 + 9 = 44 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ \text{Perimeter} = 2 \times 44 = 88 \]
### Answer: 88 meters
Expected answer: 88
Correct? Yes
Problem: Suppose the length of the rectangular playground was initially stated incorrectly and it's actually 26m, while the width remains 9m. What would be the correct perimeter of the playground in this case?
GPT-4o:
To find the correct perimeter of the rectangular playground with the updated length, we use the same formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Corrected Length (\(L\)) = 26 meters
- Width (\(W\)) = 9 meters
First, we add the corrected length and the width:
\[ L + W = 26 + 9 = 35 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ \text{Perimeter} = 2 \times 35 = 70 \]
### Answer: 70 meters
Expected answer: 70
Correct? Yes
Problem: A rectangle has a length of 9m and a width of 25m. What is the area of the rectangle?
GPT-4o:
To find the area of a rectangle, we use the formula:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Given:
- Length = 9 meters
- Width = 25 meters
Substitute the given values into the formula:
\[ \text{Area} = 9 \, \text{m} \times 25 \, \text{m} \]
\[ \text{Area} = 225 \, \text{m}^2 \]
### Answer: 225 m²
Expected answer: 225
Correct? Yes
Problem: Suppose the length of the rectangle was actually 16m and the width remained the same at 9m. What would be the new area of the rectangle in this case?
GPT-4o:
To find the new area of the rectangle, we use the same formula:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Given:
- New Length = 16 meters
- Width = 9 meters
Substitute the given values into the formula:
\[ \text{Area} = 16 \, \text{m} \times 9 \, \text{m} \]
\[ \text{Area} = 144 \, \text{m}^2 \]
### Answer: 144 m²
Expected answer: 144
Correct? Yes
Problem: A rectangle has a length of 13m and a width of 43m. What is the area of the rectangle?
GPT-4o:
To find the area of a rectangle, we use the formula:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Given:
- Length = 13 meters
- Width = 43 meters
Substitute the given values into the formula:
\[ \text{Area} = 13 \, \text{m} \times 43 \, \text{m} \]
Now, perform the multiplication:
\[ \text{Area} = 559 \, \text{m}^2 \]
### Answer: 559 m²
Expected answer: 559
Correct? Yes
Problem: Suppose we realized later that the width of the rectangle was actually 42m, not 43m. What would then be the area of the rectangle?
GPT-4o:
To find the area of the rectangle with the corrected width, we use the same formula:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Given:
- Length = 13 meters
- Width = 42 meters
Substitute the given values into the formula:
\[ \text{Area} = 13 \, \text{m} \times 42 \, \text{m} \]
Now, perform the multiplication:
\[ \text{Area} = 546 \, \text{m}^2 \]
### Answer: 546 m²
Expected answer: 546
Correct? Yes
Problem: A rectangle has a width of 27m and a length of 47m. What is the perimeter of the rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Width = 27 meters
- Length = 47 meters
First, we add the length and the width:
\[ 47 + 27 = 74 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ 2 \times 74 = 148 \]
### Answer: 148 meters
Expected answer: 148
Correct? Yes
Problem: Suppose the width of the rectangle is actually 9m instead of 27m. What would be the new perimeter of the rectangle?
GPT-4o:
To find the new perimeter of the rectangle with the updated width, we use the same formula for the perimeter of a rectangle:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Width = 9 meters
- Length = 47 meters
First, we add the length and the width:
\[ 47 + 9 = 56 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ 2 \times 56 = 112 \]
### Answer: 112 meters
Expected answer: 112
Correct? Yes
Problem: A rectangular football field has an area of 21m * 42m, calculate the area of the football field.
GPT-4o:
To find the area of a rectangular football field, we use the formula for the area of a rectangle, which is:
\[ \text{Area} = \text{length} \times \text{width} \]
Given:
- Length = 21 meters
- Width = 42 meters
Now, we calculate the area:
\[ \text{Area} = 21 \, \text{m} \times 42 \, \text{m} \]
\[ \text{Area} = 882 \, \text{m}^2 \]
### Answer: 882 m²
Expected answer: 882
Correct? Yes
Problem: A rectangle's area is found by multiplying its length by its width. If the width of a certain rectangle is 27m and the length is 25m, what is the area of the rectangle?
GPT-4o:
To find the area of the rectangle, we use the formula:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Given:
- Length = 25 meters
- Width = 27 meters
Substitute the given values into the formula:
\[ \text{Area} = 25 \, \text{m} \times 27 \, \text{m} \]
Now, perform the multiplication:
\[ \text{Area} = 25 \times 27 \]
\[ \text{Area} = 675 \, \text{m}^2 \]
### Answer: 675 m²
Expected answer: 675
Correct? Yes
Problem: Suppose the width of the rectangle in the previous problem was not 27m, but was instead 11m, with the length remaining at 25m. What would then be the new area of the rectangle?
GPT-4o:
To find the new area of the rectangle with the updated width, we use the same formula:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Given:
- Length = 25 meters
- Width = 11 meters
Substitute the given values into the formula:
\[ \text{Area} = 25 \, \text{m} \times 11 \, \text{m} \]
Now, perform the multiplication:
\[ \text{Area} = 25 \times 11 \]
\[ \text{Area} = 275 \, \text{m}^2 \]
### Answer: 275 m²
Expected answer: 275
Correct? Yes
Problem: A rectangle has a length of 45m and a width of 48m. What is the perimeter of the rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 45 meters
- Width (\(W\)) = 48 meters
First, we add the length and the width:
\[ L + W = 45 + 48 = 93 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ \text{Perimeter} = 2 \times 93 = 186 \]
### Answer: 186 meters
Expected answer: 186
Correct? Yes
Problem: In the previous example, consider if the width of the rectangle was actually 34m instead of 48m. Now, what would be the perimeter of that rectangle?
GPT-4o:
To find the perimeter of the rectangle with the updated width, we use the same formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 45 meters
- Width (\(W\)) = 34 meters
First, we add the length and the width:
\[ L + W = 45 + 34 = 79 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ \text{Perimeter} = 2 \times 79 = 158 \]
### Answer: 158 meters
Expected answer: 158
Correct? Yes
Problem: A rectangle has a length of 11 meters and a width of 7 meters. What is the perimeter of the rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length = 11 meters
- Width = 7 meters
First, we add the length and the width:
\[ 11 + 7 = 18 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ 2 \times 18 = 36 \]
### Answer: 36 meters
Expected answer: 36
Correct? Yes
Problem: Suppose the length of the rectangle was instead 10 meters while the width remained 7 meters. What would be the new perimeter of the rectangle?
GPT-4o:
To find the new perimeter of the rectangle with the updated dimensions, we use the same formula for the perimeter of a rectangle:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- New Length = 10 meters
- Width = 7 meters
First, we add the new length and the width:
\[ 10 + 7 = 17 \]
Next, we multiply the sum by 2 to find the new perimeter:
\[ 2 \times 17 = 34 \]
### Answer: 34 meters
Expected answer: 34
Correct? Yes
Problem: A rectangular field has a length of 30m and a width of 2m. Can you calculate the perimeter of the field?
GPT-4o:
To calculate the perimeter of a rectangular field, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 30 meters
- Width (\(W\)) = 2 meters
Substitute the given values into the formula:
\[ \text{Perimeter} = 2 \times (30 + 2) \]
\[ \text{Perimeter} = 2 \times 32 \]
\[ \text{Perimeter} = 64 \]
### Answer: 64 meters
Expected answer: 64
Correct? Yes
Problem: As a follow-up to the previous problem, let's say that the length of the rectangular field is reduced to 20m instead of 30m, while the width remains the same. What would the new perimeter of the field be?
GPT-4o:
To calculate the new perimeter of the rectangular field with the updated length, we use the same perimeter formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- New Length (\(L\)) = 20 meters
- Width (\(W\)) = 2 meters
Substitute the new length and the given width into the formula:
\[ \text{Perimeter} = 2 \times (20 + 2) \]
\[ \text{Perimeter} = 2 \times 22 \]
\[ \text{Perimeter} = 44 \]
### Answer: 44 meters
Expected answer: 44
Correct? Yes
Problem: A rectangle has one side that is 7m long and another side that is 8m long. What is the perimeter of the rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{length} + \text{width}) \]
Given:
- Length = 8 meters
- Width = 7 meters
Substitute the given values into the formula:
\[ \text{Perimeter} = 2 \times (8 + 7) \]
\[ \text{Perimeter} = 2 \times 15 \]
\[ \text{Perimeter} = 30 \]
### Answer: 30 meters
Expected answer: 30
Correct? Yes
Problem: Consider that the length of the rectangle was originally 19m instead of 7m. How would the new perimeter be calculated then?
GPT-4o:
Given the new dimensions of the rectangle:
- Length = 19 meters
- Width = 8 meters
We use the same formula for the perimeter of a rectangle:
\[ \text{Perimeter} = 2 \times (\text{length} + \text{width}) \]
Substitute the new values into the formula:
\[ \text{Perimeter} = 2 \times (19 + 8) \]
\[ \text{Perimeter} = 2 \times 27 \]
\[ \text{Perimeter} = 54 \]
### Answer: 54 meters
Expected answer: 54
Correct? Yes
Problem: A rectangle has a length of 14m and a width of 18m. What is the perimeter of this rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 14 meters
- Width (\(W\)) = 18 meters
First, we add the length and the width:
\[ L + W = 14 + 18 = 32 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ \text{Perimeter} = 2 \times 32 = 64 \]
### Answer: 64 meters
Expected answer: 64
Correct? Yes
Problem: Now let's consider another rectangle. The length of this rectangle is 9m rather than 14m, while the width remains the same at 18m. What is the perimeter of this new rectangle?
GPT-4o:
To find the perimeter of the new rectangle, we use the same formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 9 meters
- Width (\(W\)) = 18 meters
First, we add the length and the width:
\[ L + W = 9 + 18 = 27 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ \text{Perimeter} = 2 \times 27 = 54 \]
### Answer: 54 meters
Expected answer: 54
Correct? Yes
Problem: A rectangle has a length of 43 units and a width of 3 units. Calculate the perimeter of this rectangle.
GPT-4o:
To calculate the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length = 43 units
- Width = 3 units
First, we add the length and the width:
\[ 43 + 3 = 46 \]
Next, we multiply the sum by 2:
\[ 2 \times 46 = 92 \]
### Answer: 92 units
Expected answer: 92
Correct? Yes
Problem: Let's consider the same rectangle from before, but this time, the width has increased to 39 units while the length remained the same at 43 units. What is the new perimeter of this rectangle?
GPT-4o:
To calculate the new perimeter of the rectangle with the updated dimensions, we use the same formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length = 43 units
- Width = 39 units
First, we add the length and the width:
\[ 43 + 39 = 82 \]
Next, we multiply the sum by 2:
\[ 2 \times 82 = 164 \]
### Answer: 164 units
Expected answer: 164
Correct? Yes
Problem: A rectangle has one side that measures 36 meters and another side that measures 42 meters. What is the perimeter of this rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{length} + \text{width}) \]
Given:
- Length = 36 meters
- Width = 42 meters
First, we add the length and the width:
\[ 36 + 42 = 78 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ 2 \times 78 = 156 \]
### Answer: 156 meters
Expected answer: 156
Correct? Yes
Problem: Now suppose that the longer side of the rectangle measures 48 meters instead of 42. What would be the new perimeter of the rectangle?
GPT-4o:
To find the new perimeter of the rectangle with the updated side length, we use the same formula for the perimeter:
\[ \text{Perimeter} = 2 \times (\text{length} + \text{width}) \]
Given:
- Length = 36 meters
- New width = 48 meters
First, we add the length and the new width:
\[ 36 + 48 = 84 \]
Next, we multiply the sum by 2 to find the new perimeter:
\[ 2 \times 84 = 168 \]
### Answer: 168 meters
Expected answer: 168
Correct? Yes
Problem: A rectangle has a length of 1m and a width of 6m. What is the perimeter of this rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length = 1 meter
- Width = 6 meters
Substitute the given values into the formula:
\[ \text{Perimeter} = 2 \times (1 + 6) \]
\[ \text{Perimeter} = 2 \times 7 \]
\[ \text{Perimeter} = 14 \]
### Answer: 14 meters
Expected answer: 14
Correct? Yes
Problem: Now consider that the length of the rectangle has changed to 28m, while the width remains at 6m. What is the new perimeter of the rectangle?
GPT-4o:
To find the new perimeter of the rectangle, we use the same formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- New Length = 28 meters
- Width = 6 meters
Substitute the given values into the formula:
\[ \text{Perimeter} = 2 \times (28 + 6) \]
\[ \text{Perimeter} = 2 \times 34 \]
\[ \text{Perimeter} = 68 \]
### Answer: 68 meters
Expected answer: 68
Correct? Yes
Problem: A rectangle has two sides measuring 45m and two sides measuring 8m. What is the perimeter of this rectangle?
GPT-4o:
To find the perimeter of a rectangle, we add up the lengths of all four sides. A rectangle has two pairs of equal sides. In this case, the sides measure 45 meters and 8 meters.
The formula for the perimeter \( P \) of a rectangle is:
\[ P = 2 \times (\text{length} + \text{width}) \]
Given:
- Length = 45 meters
- Width = 8 meters
Substitute these values into the formula:
\[ P = 2 \times (45 + 8) \]
\[ P = 2 \times 53 \]
\[ P = 106 \]
### Answer: 106 meters
Expected answer: 106
Correct? Yes
Problem: A rectangular garden has a length of 25 meters and a width of 37 meters. What is the perimeter of the garden?
GPT-4o:
To find the perimeter of a rectangular garden, we use the formula for the perimeter of a rectangle, which is given by:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 25 meters
- Width (\(W\)) = 37 meters
First, we add the length and the width:
\[ L + W = 25 + 37 = 62 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ \text{Perimeter} = 2 \times 62 = 124 \]
### Answer: 124 meters
Expected answer: 124
Correct? Yes
Problem: Suppose we reduce the length of the garden from 25 meters to 16 meters, while the width remains unchanged at 37 meters. What would be the new perimeter of the garden?
GPT-4o:
To find the new perimeter of the rectangular garden after reducing the length, we use the same perimeter formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given the new dimensions:
- New Length (\(L\)) = 16 meters
- Width (\(W\)) = 37 meters
First, we add the new length and the width:
\[ L + W = 16 + 37 = 53 \]
Next, we multiply the sum by 2 to find the new perimeter:
\[ \text{Perimeter} = 2 \times 53 = 106 \]
### Answer: 106 meters
Expected answer: 106
Correct? Yes
Problem: A rectangle has length 36m and width 3m. What is the perimeter of this rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 36 meters
- Width (\(W\)) = 3 meters
First, we add the length and the width:
\[ L + W = 36 + 3 = 39 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ \text{Perimeter} = 2 \times 39 = 78 \]
### Answer: 78 meters
Expected answer: 78
Correct? Yes
Problem: Suppose we made a change to the rectangle from the previous problem. Instead of a length of 36m, we shortened it to only 1m. What would be the new perimeter of this rectangle?
GPT-4o:
To find the new perimeter of the rectangle with the updated dimensions, we use the same perimeter formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given the new dimensions:
- Length (\(L\)) = 1 meter
- Width (\(W\)) = 3 meters
First, we add the new length and the width:
\[ L + W = 1 + 3 = 4 \]
Next, we multiply the sum by 2 to find the new perimeter:
\[ \text{Perimeter} = 2 \times 4 = 8 \]
### Answer: 8 meters
Expected answer: 8
Correct? Yes
Problem: A rectangular garden has a length of 45 meters and a width of 50 meters. Calculate the perimeter of the garden.
GPT-4o:
To find the perimeter of a rectangular garden, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 45 meters
- Width (\(W\)) = 50 meters
Substitute the given values into the formula:
\[ \text{Perimeter} = 2 \times (45 + 50) \]
\[ \text{Perimeter} = 2 \times 95 \]
\[ \text{Perimeter} = 190 \]
### Answer: 190 meters
Expected answer: 190
Correct? Yes
Problem: A rectangle has a length of 44m and a width of 34m. What is the perimeter of this rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 44 meters
- Width (\(W\)) = 34 meters
First, we add the length and the width:
\[ L + W = 44 + 34 = 78 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ \text{Perimeter} = 2 \times 78 = 156 \]
### Answer: 156 meters
Expected answer: 156
Correct? Yes
Problem: Now suppose the width of the rectangle changes to 36m while the length remains the same at 44m. What would be the new perimeter of the rectangle in this case?
GPT-4o:
To find the new perimeter of the rectangle with the updated width, we use the same perimeter formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 44 meters
- New Width (\(W\)) = 36 meters
First, we add the length and the new width:
\[ L + W = 44 + 36 = 80 \]
Next, we multiply the sum by 2 to find the new perimeter:
\[ \text{Perimeter} = 2 \times 80 = 160 \]
### Answer: 160 meters
Expected answer: 160
Correct? Yes
Problem: A rectangle has a length of 42 m and a width of 29 m. Calculate the perimeter of the rectangle.
GPT-4o:
To calculate the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 42 meters
- Width (\(W\)) = 29 meters
First, we add the length and the width:
\[ L + W = 42 + 29 = 71 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ \text{Perimeter} = 2 \times 71 = 142 \]
### Answer: 142 meters
Expected answer: 142
Correct? Yes
Problem: Suppose the width of the rectangle was actually 38 m instead of 29 m. What would then be the new perimeter of the rectangle?
GPT-4o:
To calculate the new perimeter of the rectangle with the updated width, we use the same formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 42 meters
- New Width (\(W\)) = 38 meters
First, we add the length and the new width:
\[ L + W = 42 + 38 = 80 \]
Next, we multiply the sum by 2 to find the new perimeter:
\[ \text{Perimeter} = 2 \times 80 = 160 \]
### Answer: 160 meters
Expected answer: 160
Correct? Yes
Problem: A rectangle has a length of 44 meters and a width of 35 meters. What is the perimeter of the rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 44 meters
- Width (\(W\)) = 35 meters
First, we add the length and the width:
\[ L + W = 44 + 35 = 79 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ \text{Perimeter} = 2 \times 79 = 158 \]
### Answer: 158 meters
Expected answer: 158
Correct? Yes
Problem: Consider the same rectangle from the previous problem, but now the width has changed to 43 meters while the length remains the same at 44 meters. What is the new perimeter of the rectangle?
GPT-4o:
To find the new perimeter of the rectangle with the updated width, we use the same perimeter formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 44 meters
- New Width (\(W\)) = 43 meters
First, we add the length and the new width:
\[ L + W = 44 + 43 = 87 \]
Next, we multiply the sum by 2 to find the new perimeter:
\[ \text{Perimeter} = 2 \times 87 = 174 \]
### Answer: 174 meters
Expected answer: 174
Correct? Yes
Problem: A rectangle has a length of 44m and a width of 45m. What is the perimeter of this rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\( L \)) = 44 meters
- Width (\( W \)) = 45 meters
First, we add the length and the width:
\[ L + W = 44 + 45 = 89 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ \text{Perimeter} = 2 \times 89 = 178 \]
### Answer: 178 meters
Expected answer: 178
Correct? Yes
Problem: The previous rectangle had its width increased from 45m to 48m while keeping the length the same. What is the new perimeter of the rectangle?
GPT-4o:
To find the new perimeter of the rectangle after increasing the width, we use the same perimeter formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\( L \)) = 44 meters (unchanged)
- New Width (\( W \)) = 48 meters
First, we add the new width to the length:
\[ L + W = 44 + 48 = 92 \]
Next, we multiply the sum by 2 to find the new perimeter:
\[ \text{Perimeter} = 2 \times 92 = 184 \]
### Answer: 184 meters
Expected answer: 184
Correct? Yes
Problem: A rectangle has a length of 3 meters and a width of 39 meters. What is the perimeter of this rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length = 3 meters
- Width = 39 meters
First, we add the length and the width:
\[ \text{Length} + \text{Width} = 3 + 39 = 42 \text{ meters} \]
Next, we multiply the sum by 2 to find the perimeter:
\[ \text{Perimeter} = 2 \times 42 = 84 \text{ meters} \]
### Answer: 84 meters
Expected answer: 84
Correct? Yes
Problem: Suppose the length of the rectangle was actually 17 meters instead of 3 meters, keeping the width the same. What would be the new perimeter of the rectangle now?
GPT-4o:
To find the new perimeter of the rectangle with the updated length, we use the same formula for the perimeter of a rectangle:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- New Length = 17 meters
- Width = 39 meters
First, we add the new length and the width:
\[ \text{Length} + \text{Width} = 17 + 39 = 56 \text{ meters} \]
Next, we multiply the sum by 2 to find the new perimeter:
\[ \text{Perimeter} = 2 \times 56 = 112 \text{ meters} \]
### Answer: 112 meters
Expected answer: 112
Correct? Yes
Problem: A rectangular garden has sides that measure 32m and 12m. What is the perimeter of this garden?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{length} + \text{width}) \]
Given the length \( l = 32 \) meters and the width \( w = 12 \) meters, we can substitute these values into the formula:
\[ \text{Perimeter} = 2 \times (32 + 12) \]
First, calculate the sum inside the parentheses:
\[ 32 + 12 = 44 \]
Next, multiply by 2:
\[ 2 \times 44 = 88 \]
### Answer: 88 meters
Expected answer: 88
Correct? Yes
Problem: A rectangular swimming pool has a length of 24 feet and a width of 6 feet. What is the perimeter of the swimming pool?
GPT-4o:
To find the perimeter of a rectangular swimming pool, we use the formula for the perimeter of a rectangle, which is:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length = 24 feet
- Width = 6 feet
First, we add the length and the width:
\[ 24 + 6 = 30 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ 2 \times 30 = 60 \]
### Answer: 60 feet
Expected answer: 60
Correct? Yes
Problem: A rectangle has a length of 27m and a width of 21m. What is the area of this rectangle?
GPT-4o:
To find the area of a rectangle, we use the formula:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Given:
- Length = 27 meters
- Width = 21 meters
Substitute the given values into the formula:
\[ \text{Area} = 27 \, \text{m} \times 21 \, \text{m} \]
Now, perform the multiplication:
\[ \text{Area} = 27 \times 21 = 567 \, \text{m}^2 \]
### Answer: 567 m²
Expected answer: 567
Correct? Yes
Problem: In the previous problem, the rectangle had a length of 27m and a width of 21m. Now, consider that the width of the rectangle is instead 8m. What is the area of this rectangle?
GPT-4o:
To find the area of the rectangle with the new dimensions, we use the same formula:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Given:
- Length = 27 meters
- Width = 8 meters
Substitute the given values into the formula:
\[ \text{Area} = 27 \, \text{m} \times 8 \, \text{m} \]
Now, perform the multiplication:
\[ \text{Area} = 27 \times 8 = 216 \, \text{m}^2 \]
### Answer: 216 m²
Expected answer: 216
Correct? Yes
Problem: A rectangle has twice the length of a square. If the square's side is 49m, what is the length of the rectangle?
GPT-4o:
To find the length of the rectangle, we first need to determine the side length of the square. According to the problem, the side length of the square is 49 meters.
Since the rectangle has twice the length of the square, we multiply the side length of the square by 2:
\[ \text{Length of the rectangle} = 2 \times \text{side length of the square} \]
Substituting the given side length of the square:
\[ \text{Length of the rectangle} = 2 \times 49 \]
\[ \text{Length of the rectangle} = 98 \]
### Answer: 98 meters
Expected answer: 98
Correct? Yes
Problem: In the previous problem, we found the length of the rectangle when the square's side was 49m. Now, consider a square with a side length of 18m. What would be the length of the rectangle if it's still twice the length of the square?
GPT-4o:
To find the length of the rectangle when the side length of the square is 18 meters, we use the same approach as before. The rectangle's length is twice the length of the square's side.
First, we identify the side length of the square, which is 18 meters.
Next, we calculate the length of the rectangle by multiplying the side length of the square by 2:
\[ \text{Length of the rectangle} = 2 \times \text{side length of the square} \]
Substituting the given side length of the square:
\[ \text{Length of the rectangle} = 2 \times 18 \]
\[ \text{Length of the rectangle} = 36 \]
### Answer: 36 meters
Expected answer: 36
Correct? Yes
Problem: A rectangle has a length of 50m and a width of 36m. What is the perimeter of this rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 50 meters
- Width (\(W\)) = 36 meters
First, we add the length and the width:
\[ L + W = 50 + 36 = 86 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ \text{Perimeter} = 2 \times 86 = 172 \]
### Answer: 172 meters
Expected answer: 172
Correct? Yes
Problem: In our previous rectangle, if we now increase the width from 36m to 44m, while keeping the length the same, what will be the new perimeter of the rectangle?
GPT-4o:
To find the new perimeter of the rectangle after increasing the width, we will use the same perimeter formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 50 meters (unchanged)
- New Width (\(W\)) = 44 meters
First, we add the length and the new width:
\[ L + W = 50 + 44 = 94 \]
Next, we multiply the sum by 2 to find the new perimeter:
\[ \text{Perimeter} = 2 \times 94 = 188 \]
### Answer: 188 meters
Expected answer: 188
Correct? Yes
Problem: A rectangle has a length of 8m and a width of 44m. What is the perimeter of the rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 8 meters
- Width (\(W\)) = 44 meters
Substitute the given values into the formula:
\[ \text{Perimeter} = 2 \times (8 + 44) \]
\[ \text{Perimeter} = 2 \times 52 \]
\[ \text{Perimeter} = 104 \]
### Answer: 104 meters
Expected answer: 104
Correct? Yes
Problem: Consider the same rectangle from the previous problem, but now its length increases to 40m, while the width remains the same at 44m. What is the new perimeter of the rectangle?
GPT-4o:
To find the new perimeter of the rectangle after the length increases, we use the same perimeter formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- New Length (\(L\)) = 40 meters
- Width (\(W\)) = 44 meters
Substitute the new length and the given width into the formula:
\[ \text{Perimeter} = 2 \times (40 + 44) \]
\[ \text{Perimeter} = 2 \times 84 \]
\[ \text{Perimeter} = 168 \]
### Answer: 168 meters
Expected answer: 168
Correct? Yes
Problem: If a rectangle has a length of 47 feet and a width of 21 feet, what is the area of the rectangle?
GPT-4o:
To find the area of a rectangle, we use the formula:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Given:
- Length = 47 feet
- Width = 21 feet
Substitute the given values into the formula:
\[ \text{Area} = 47 \, \text{feet} \times 21 \, \text{feet} \]
Now, perform the multiplication:
\[ 47 \times 21 = 987 \]
So, the area of the rectangle is 987 square feet.
### Answer: 987 square feet.
Expected answer: 987
Correct? Yes
Problem: Now, suppose we adjusted the measurements of the previous rectangle. If the length of the rectangle remained the same at 21 feet, but the width of the rectangle was reduced to 15 feet, what would the new area of the rectangle be?
GPT-4o:
To find the new area of the rectangle, we use the same formula:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Given:
- Length = 21 feet
- Width = 15 feet
Substitute the given values into the formula:
\[ \text{Area} = 21 \, \text{feet} \times 15 \, \text{feet} \]
Now, perform the multiplication:
\[ 21 \times 15 = 315 \]
So, the new area of the rectangle is 315 square feet.
### Answer: 315 square feet.
Expected answer: 315
Correct? Yes
Problem: A rectangle has a length of 19m and a breadth of 28m. What is the perimeter of the rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Breadth}) \]
Given:
- Length (\(L\)) = 19 meters
- Breadth (\(B\)) = 28 meters
First, we add the length and the breadth:
\[ L + B = 19 + 28 = 47 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ \text{Perimeter} = 2 \times 47 = 94 \]
### Answer: 94 meters
Expected answer: 94
Correct? Yes
Problem: In the previous problem, what if the length of the rectangle was 27m instead of 19m while keeping the breadth the same? What would be the new perimeter?
GPT-4o:
To find the new perimeter of the rectangle with the updated length, we use the same formula for the perimeter of a rectangle:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Breadth}) \]
Given:
- New Length (\(L\)) = 27 meters
- Breadth (\(B\)) = 28 meters
First, we add the new length and the breadth:
\[ L + B = 27 + 28 = 55 \]
Next, we multiply the sum by 2 to find the new perimeter:
\[ \text{Perimeter} = 2 \times 55 = 110 \]
### Answer: 110 meters
Expected answer: 110
Correct? Yes
Problem: One side of a rectangle is four times the length of the other side. If the longer side measures 39 meters, what is the area of the rectangle when the shorter side is 10 meters?
GPT-4o:
Let's first verify the given information and then calculate the area of the rectangle.
We are given:
- The longer side of the rectangle is 39 meters.
- The shorter side of the rectangle is 10 meters.
The area \( A \) of a rectangle is calculated using the formula:
\[ A = \text{length} \times \text{width} \]
Here, the length is 39 meters and the width is 10 meters. Therefore, the area is:
\[ A = 39 \times 10 \]
\[ A = 390 \]
### Answer: 390 square meters.
Expected answer: 390
Correct? Yes
Problem: Suppose we discovered an error in the original measurements of the rectangle. The longer side is actually 37 meters instead of 39 meters. With the shorter side still at 10 meters, what is the new area of the rectangle?
GPT-4o:
To find the new area of the rectangle with the corrected measurements, we will use the same formula for the area of a rectangle:
\[ A = \text{length} \times \text{width} \]
Given:
- The corrected longer side (length) is 37 meters.
- The shorter side (width) is still 10 meters.
Now, we calculate the area:
\[ A = 37 \times 10 \]
\[ A = 370 \]
### Answer: 370 square meters.
Expected answer: 370
Correct? Yes
Problem: A rectangle has identical sides, each measuring 19 meters. What is the total perimeter of the rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{length} + \text{width}) \]
Given that the rectangle has identical sides, each measuring 19 meters, it means the length and the width are both 19 meters.
Substituting the values into the formula:
\[ \text{Perimeter} = 2 \times (19 + 19) \]
\[ \text{Perimeter} = 2 \times 38 \]
\[ \text{Perimeter} = 76 \]
### Answer: 76 meters
Expected answer: 76
Correct? Yes
Problem: The rectangle mentioned in the previous problem has been resized. The new rectangle now has a length of 19 meters but a width of 6 meters. What is the new total perimeter of the rectangle?
GPT-4o:
To find the perimeter of the new rectangle, we use the same formula:
\[ \text{Perimeter} = 2 \times (\text{length} + \text{width}) \]
Given the new dimensions:
- Length = 19 meters
- Width = 6 meters
Substituting these values into the formula:
\[ \text{Perimeter} = 2 \times (19 + 6) \]
\[ \text{Perimeter} = 2 \times 25 \]
\[ \text{Perimeter} = 50 \]
### Answer: 50 meters
Expected answer: 50
Correct? Yes
Problem: The area of a rectangle is given by the equation length * width. If the length is 9m and the width is 26m, what is the area of the rectangle?
GPT-4o:
To find the area of the rectangle, we use the formula:
\[ \text{Area} = \text{length} \times \text{width} \]
Given:
- Length = 9 meters
- Width = 26 meters
Substitute the given values into the formula:
\[ \text{Area} = 9 \, \text{m} \times 26 \, \text{m} \]
Now, perform the multiplication:
\[ \text{Area} = 234 \, \text{m}^2 \]
### Answer: 234 m²
Expected answer: 234
Correct? Yes
Problem: The rectangle now has a new dimension with its length increased to 47m while the width is still 9m. What is the new area of the rectangle?
GPT-4o:
To find the new area of the rectangle with the updated dimensions, we use the same formula:
\[ \text{Area} = \text{length} \times \text{width} \]
Given:
- New Length = 47 meters
- Width = 9 meters
Substitute the given values into the formula:
\[ \text{Area} = 47 \, \text{m} \times 9 \, \text{m} \]
Now, perform the multiplication:
\[ \text{Area} = 423 \, \text{m}^2 \]
### Answer: 423 m²
Expected answer: 423
Correct? Yes
Problem: A rectangle has one side measuring 19cm and another side measuring 7cm. What is the perimeter of the rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{length} + \text{width}) \]
Given:
- Length = 19 cm
- Width = 7 cm
First, we add the length and the width:
\[ 19 \, \text{cm} + 7 \, \text{cm} = 26 \, \text{cm} \]
Next, we multiply the sum by 2 to find the perimeter:
\[ 2 \times 26 \, \text{cm} = 52 \, \text{cm} \]
### Answer: 52 cm
Expected answer: 52
Correct? Yes
Problem: Suppose we extended the rectangle's length from the previous question from 19cm to 21cm. In this case, what would be the new perimeter?
GPT-4o:
To find the new perimeter of the rectangle after extending the length, we use the same perimeter formula:
\[ \text{Perimeter} = 2 \times (\text{length} + \text{width}) \]
Given the new dimensions:
- New Length = 21 cm
- Width = 7 cm
First, we add the new length and the width:
\[ 21 \, \text{cm} + 7 \, \text{cm} = 28 \, \text{cm} \]
Next, we multiply the sum by 2 to find the new perimeter:
\[ 2 \times 28 \, \text{cm} = 56 \, \text{cm} \]
### Answer: 56 cm
Expected answer: 56
Correct? Yes
Problem: A shutter is in the shape of a rectangle. Its length is 10m and its width is 6m. What is the total area of the shutter?
GPT-4o:
To find the area of a rectangle, we use the formula:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Given:
- Length = 10 meters
- Width = 6 meters
Substituting the given values into the formula:
\[ \text{Area} = 10 \, \text{m} \times 6 \, \text{m} \]
\[ \text{Area} = 60 \, \text{m}^2 \]
### Answer: 60 m²
Expected answer: 60
Correct? Yes
Problem: A rectangle has a length of 24 units and a width of 15 units. What is the perimeter of the rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length = 24 units
- Width = 15 units
First, we add the length and the width:
\[ 24 + 15 = 39 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ 2 \times 39 = 78 \]
### Answer: 78 units
Expected answer: 78
Correct? Yes
Problem: Now, imagine the rectangle's width was reduced to 10 units, while the length remained 24 units. What would be the new perimeter of the rectangle under these conditions?
GPT-4o:
To find the new perimeter of the rectangle with the updated dimensions, we use the same perimeter formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given the new dimensions:
- Length = 24 units
- Width = 10 units
First, we add the length and the width:
\[ 24 + 10 = 34 \]
Next, we multiply the sum by 2 to find the new perimeter:
\[ 2 \times 34 = 68 \]
### Answer: 68 units
Expected answer: 68
Correct? Yes
Problem: If a rectangular plot of land has a length of 32m and a width of 27m, what is the area of the plot in square meters?
GPT-4o:
To find the area of a rectangular plot of land, we use the formula for the area of a rectangle, which is:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Given:
- Length = 32 meters
- Width = 27 meters
Substitute the given values into the formula:
\[ \text{Area} = 32 \, \text{m} \times 27 \, \text{m} \]
Now, perform the multiplication:
\[ \text{Area} = 32 \times 27 \]
To simplify the multiplication, we can break it down:
\[ 32 \times 27 = 32 \times (20 + 7) \]
\[ = 32 \times 20 + 32 \times 7 \]
\[ = 640 + 224 \]
\[ = 864 \]
So, the area of the rectangular plot of land is:
### Answer: 864 square meters.
Expected answer: 864
Correct? Yes
Problem: In the previous scenario, suppose the width of the rectangular plot of land was actually 18m rather than 27m. What would then be the area of the plot in square meters?
GPT-4o:
To find the area of the rectangular plot of land with the updated width, we use the same formula for the area of a rectangle:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Given:
- Length = 32 meters
- Width = 18 meters
Substitute the given values into the formula:
\[ \text{Area} = 32 \, \text{m} \times 18 \, \text{m} \]
Now, perform the multiplication:
\[ \text{Area} = 32 \times 18 \]
To simplify the multiplication, we can break it down:
\[ 32 \times 18 = 32 \times (10 + 8) \]
\[ = 32 \times 10 + 32 \times 8 \]
\[ = 320 + 256 \]
\[ = 576 \]
So, the area of the rectangular plot of land with the updated width is:
### Answer: 576 square meters.
Expected answer: 576
Correct? Yes
Problem: A rectangle has one side that measures 15m and another side that measures 10m. What is the perimeter of this rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{length} + \text{width}) \]
Given:
- Length = 15 meters
- Width = 10 meters
First, we add the length and the width:
\[ 15 + 10 = 25 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ 2 \times 25 = 50 \]
### Answer: 50 meters
Expected answer: 50
Correct? Yes
Problem: Now consider the rectangle has one side that measures 33m instead of 15m while the other side still measures 10m. What is the new perimeter of this rectangle?
GPT-4o:
To find the new perimeter of the rectangle, we use the same formula:
\[ \text{Perimeter} = 2 \times (\text{length} + \text{width}) \]
Given:
- Length = 33 meters
- Width = 10 meters
First, we add the length and the width:
\[ 33 + 10 = 43 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ 2 \times 43 = 86 \]
### Answer: 86 meters
Expected answer: 86
Correct? Yes
Problem: A rectangle has a length of 44m and a width of 37m. What is the perimeter of this rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\( L \)) = 44 meters
- Width (\( W \)) = 37 meters
First, we add the length and the width:
\[ L + W = 44 + 37 = 81 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ \text{Perimeter} = 2 \times 81 = 162 \]
### Answer: 162 meters
Expected answer: 162
Correct? Yes
Problem: What would the perimeter of the rectangle be if the length was 25m instead of 44m but the width remained the same at 37m?
GPT-4o:
To find the perimeter of the rectangle with the new dimensions, we use the same formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- New Length (\( L \)) = 25 meters
- Width (\( W \)) = 37 meters
First, we add the new length and the width:
\[ L + W = 25 + 37 = 62 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ \text{Perimeter} = 2 \times 62 = 124 \]
### Answer: 124 meters
Expected answer: 124
Correct? Yes
Problem: A rectangle has length of 14 cm and width 8 cm. What is the perimeter of the rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length = 14 cm
- Width = 8 cm
First, we add the length and the width:
\[ 14 \, \text{cm} + 8 \, \text{cm} = 22 \, \text{cm} \]
Next, we multiply the sum by 2 to find the perimeter:
\[ 2 \times 22 \, \text{cm} = 44 \, \text{cm} \]
### Answer: 44 cm
Expected answer: 44
Correct? Yes
Problem: If the length of the rectangle remains 14 cm, but the width is increased to 18 cm, what would be the new perimeter of the rectangle now?
GPT-4o:
To find the new perimeter of the rectangle with the updated dimensions, we use the same formula for the perimeter of a rectangle:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length = 14 cm
- Width = 18 cm
First, we add the length and the width:
\[ 14 \, \text{cm} + 18 \, \text{cm} = 32 \, \text{cm} \]
Next, we multiply the sum by 2 to find the new perimeter:
\[ 2 \times 32 \, \text{cm} = 64 \, \text{cm} \]
### Answer: 64 cm
Expected answer: 64
Correct? Yes
Problem: A rectangle has a length of 26m and a width of 28m. What is the perimeter of this rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 26 meters
- Width (\(W\)) = 28 meters
First, we add the length and the width:
\[ L + W = 26 + 28 = 54 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ \text{Perimeter} = 2 \times 54 = 108 \]
### Answer: 108 meters
Expected answer: 108
Correct? Yes
Problem: In the previous scenario, if the rectangle was actually a square with all sides measuring 26m, what would the new perimeter be?
GPT-4o:
For a square, all four sides are of equal length. The perimeter of a square is calculated using the formula:
\[ \text{Perimeter} = 4 \times \text{Side length} \]
Given:
- Side length = 26 meters
We multiply the side length by 4 to find the perimeter:
\[ \text{Perimeter} = 4 \times 26 = 104 \]
### Answer: 104 meters
Expected answer: 104
Correct? Yes
Problem: A rectangle has a length of 15m and a width of 37m. What is the perimeter of this rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 15 meters
- Width (\(W\)) = 37 meters
First, we add the length and the width:
\[ L + W = 15 + 37 = 52 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ \text{Perimeter} = 2 \times 52 = 104 \]
### Answer: 104 meters
Expected answer: 104
Correct? Yes
Problem: The width of the rectangle mentioned in the previous problem is now changed to 19m, while the length is still 15m. What is the new perimeter of this rectangle?
GPT-4o:
To find the new perimeter of the rectangle with the updated width, we use the same perimeter formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 15 meters
- New Width (\(W\)) = 19 meters
First, we add the length and the new width:
\[ L + W = 15 + 19 = 34 \]
Next, we multiply the sum by 2 to find the new perimeter:
\[ \text{Perimeter} = 2 \times 34 = 68 \]
### Answer: 68 meters
Expected answer: 68
Correct? Yes
Problem: A rectangle has a length of 42m and a width of 40m. What is the perimeter of this rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 42 meters
- Width (\(W\)) = 40 meters
First, we add the length and the width:
\[ L + W = 42 + 40 = 82 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ \text{Perimeter} = 2 \times 82 = 164 \]
### Answer: 164 meters
Expected answer: 164
Correct? Yes
Problem: The rectangle you were previously given now has a length of 42m and the width has been adjusted to 24m. What is the new perimeter of this rectangle?
GPT-4o:
To find the new perimeter of the rectangle, we use the same formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 42 meters
- Width (\(W\)) = 24 meters
First, we add the length and the width:
\[ L + W = 42 + 24 = 66 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ \text{Perimeter} = 2 \times 66 = 132 \]
### Answer: 132 meters
Expected answer: 132
Correct? Yes
Problem: A rectangular garden has one side that measures 21 feet and another side that measures 30 feet. What is the perimeter of the garden?
GPT-4o:
To find the perimeter of a rectangular garden, we use the formula for the perimeter of a rectangle, which is given by:
\[ \text{Perimeter} = 2 \times (\text{length} + \text{width}) \]
In this case, the length of the garden is 30 feet and the width is 21 feet. Plugging these values into the formula, we get:
\[ \text{Perimeter} = 2 \times (30 \text{ feet} + 21 \text{ feet}) \]
First, we add the length and the width:
\[ 30 \text{ feet} + 21 \text{ feet} = 51 \text{ feet} \]
Next, we multiply by 2:
\[ 2 \times 51 \text{ feet} = 102 \text{ feet} \]
### Answer: 102 feet
Expected answer: 102
Correct? Yes
Problem: Now, imagine that the side of the garden that initially measured 30 feet was actually only 5 feet long. What would be the new perimeter of the garden?
GPT-4o:
To find the new perimeter of the rectangular garden with the updated dimensions, we use the same formula for the perimeter of a rectangle:
\[ \text{Perimeter} = 2 \times (\text{length} + \text{width}) \]
In this case, the new length of the garden is 5 feet and the width remains 21 feet. Plugging these values into the formula, we get:
\[ \text{Perimeter} = 2 \times (5 \text{ feet} + 21 \text{ feet}) \]
First, we add the length and the width:
\[ 5 \text{ feet} + 21 \text{ feet} = 26 \text{ feet} \]
Next, we multiply by 2:
\[ 2 \times 26 \text{ feet} = 52 \text{ feet} \]
### Answer: 52 feet
Expected answer: 52
Correct? Yes
Problem: A certain playground is in the shape of a rectangle. The area of the playground is given as the product of 17 and 44. How many square meters is the playground?
GPT-4o:
To find the area of the playground, we need to multiply the given dimensions, 17 and 44. First, we perform the multiplication: \[ 17 \times 44 \] We can break this down using the distributive property: \[ 17 \times 44 = 17 \times (40 + 4) \] \[ = (17 \times 40) + (17 \times 4) \] Now, calculate each part separately: \[ 17 \times 40 = 680 \] \[ 17 \times 4 = 68 \] Add the two results together: \[ 680 + 68 = 748 \] Therefore, the area of the playground is: ### Answer: 748 square meters.
Expected answer: 748
Correct? Yes
Problem: Let's assume that the length of the playground was mistakenly recorded. Instead of 44m, the correct measurement is 37m. Given this information, how many square meters is the playground now?
GPT-4o:
To find the correct area of the playground with the updated length, we need to multiply the new dimensions, 17 meters and 37 meters. First, we perform the multiplication: \[ 17 \times 37 \] We can break this down using the distributive property: \[ 17 \times 37 = 17 \times (30 + 7) \] \[ = (17 \times 30) + (17 \times 7) \] Now, calculate each part separately: \[ 17 \times 30 = 510 \] \[ 17 \times 7 = 119 \] Add the two results together: \[ 510 + 119 = 629 \] Therefore, the correct area of the playground is: ### Answer: 629 square meters.
Expected answer: 629
Correct? Yes
Problem: Rectangle ABCD has a width of 47m and a length of 50m. What is the perimeter of the rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Width (\( W \)) = 47 meters
- Length (\( L \)) = 50 meters
Substitute the given values into the formula:
\[ \text{Perimeter} = 2 \times (50 + 47) \]
\[ \text{Perimeter} = 2 \times 97 \]
\[ \text{Perimeter} = 194 \]
### Answer: 194 meters
Expected answer: 194
Correct? Yes
Problem: Now consider that the rectangle ABCD's dimensions have changed slightly. The width is still 47m, but the length has adjusted to 48m. What is the new perimeter of the rectangle?
GPT-4o:
To find the new perimeter of the rectangle with the updated dimensions, we use the same perimeter formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Width (\( W \)) = 47 meters
- Length (\( L \)) = 48 meters
Substitute the new values into the formula:
\[ \text{Perimeter} = 2 \times (48 + 47) \]
\[ \text{Perimeter} = 2 \times 95 \]
\[ \text{Perimeter} = 190 \]
### Answer: 190 meters
Expected answer: 190
Correct? Yes
Problem: A rectangle has a length of 11m and a width of 31m. What is the perimeter of the rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 11 meters
- Width (\(W\)) = 31 meters
First, we add the length and the width:
\[ L + W = 11 + 31 = 42 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ \text{Perimeter} = 2 \times 42 = 84 \]
### Answer: 84 meters
Expected answer: 84
Correct? Yes
Problem: Suppose we later find out that the actual length of the rectangle is 36m, not 11m as we originally thought, while the width remains the same at 31m. What would then be the new perimeter of the rectangle?
GPT-4o:
To find the new perimeter of the rectangle with the updated length, we use the same formula for the perimeter of a rectangle:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- New Length (\(L\)) = 36 meters
- Width (\(W\)) = 31 meters
First, we add the new length and the width:
\[ L + W = 36 + 31 = 67 \]
Next, we multiply the sum by 2 to find the new perimeter:
\[ \text{Perimeter} = 2 \times 67 = 134 \]
### Answer: 134 meters
Expected answer: 134
Correct? Yes
Problem: A rectangular garden has a length of 39m and a width of 19m. What is the area of this garden?
GPT-4o:
To find the area of a rectangular garden, we use the formula:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Given:
- Length = 39 meters
- Width = 19 meters
Substitute the given values into the formula:
\[ \text{Area} = 39 \, \text{m} \times 19 \, \text{m} \]
Now, perform the multiplication:
\[ 39 \times 19 = 741 \]
So, the area of the garden is 741 square meters.
### Answer: 741 square meters
Expected answer: 741
Correct? Yes
Problem: If the length of the rectangular garden was actually 35m instead of 39m, keeping the width the same, what would be the new area of the garden?
GPT-4o:
To find the new area of the rectangular garden with the updated length, we use the same area formula:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Given:
- New Length = 35 meters
- Width = 19 meters
Substitute the new length and the given width into the formula:
\[ \text{Area} = 35 \, \text{m} \times 19 \, \text{m} \]
Now, perform the multiplication:
\[ 35 \times 19 = 665 \]
So, the new area of the garden is 665 square meters.
### Answer: 665 square meters
Expected answer: 665
Correct? Yes
Problem: A carpenter wants to install a wooden floor in a rectangular hallway. The wooden plank has a width of 29 cm and the carpenter has cut it to a length of 11 cm. How many square centimeters of area does one cut wooden plank cover?
GPT-4o:
To find the area of one cut wooden plank, we need to multiply its width by its length.
Given:
- Width of the plank = 29 cm
- Length of the plank = 11 cm
The area \( A \) of the plank is calculated as follows:
\[ A = \text{width} \times \text{length} \]
\[ A = 29 \, \text{cm} \times 11 \, \text{cm} \]
\[ A = 319 \, \text{cm}^2 \]
### Answer: 319 cm²
Expected answer: 319
Correct? Yes
Problem: If instead the carpenter decided to cut the same 29 cm wide wooden plank into shorter lengths of 5 cm each, how many square centimeters of area does one of these shorter cut wooden planks cover?
GPT-4o:
To find the area of one of the shorter cut wooden planks, we need to multiply its width by its new length.
Given:
- Width of the plank = 29 cm
- New length of the plank = 5 cm
The area \( A \) of the shorter plank is calculated as follows:
\[ A = \text{width} \times \text{length} \]
\[ A = 29 \, \text{cm} \times 5 \, \text{cm} \]
\[ A = 145 \, \text{cm}^2 \]
### Answer: 145 cm²
Expected answer: 145
Correct? Yes
Problem: A rectangle has a length of 25m and a width of 11m. What is the perimeter of the rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 25 meters
- Width (\(W\)) = 11 meters
Substitute the given values into the formula:
\[ \text{Perimeter} = 2 \times (25 + 11) \]
\[ \text{Perimeter} = 2 \times 36 \]
\[ \text{Perimeter} = 72 \]
### Answer: 72 meters
Expected answer: 72
Correct? Yes
Problem: Now, suppose the rectangle's width is changed from 11m to 17m. What will be the new perimeter of the rectangle?
GPT-4o:
To find the new perimeter of the rectangle with the updated width, we use the same perimeter formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 25 meters
- New Width (\(W\)) = 17 meters
Substitute the new width into the formula:
\[ \text{Perimeter} = 2 \times (25 + 17) \]
\[ \text{Perimeter} = 2 \times 42 \]
\[ \text{Perimeter} = 84 \]
### Answer: 84 meters
Expected answer: 84
Correct? Yes
Problem: A rectangle has a length of 49m and a breadth of 9m. What is the perimeter of the rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{length} + \text{breadth}) \]
Given:
- Length (\( l \)) = 49 meters
- Breadth (\( b \)) = 9 meters
Substitute the given values into the formula:
\[ \text{Perimeter} = 2 \times (49 + 9) \]
\[ \text{Perimeter} = 2 \times 58 \]
\[ \text{Perimeter} = 116 \]
### Answer: 116 meters
Expected answer: 116
Correct? Yes
Problem: Suppose that the breadth of the same rectangle was instead 19m, while the length remains the same at 49m. What would be the new perimeter of the rectangle?
GPT-4o:
To find the new perimeter of the rectangle with the updated breadth, we use the same perimeter formula:
\[ \text{Perimeter} = 2 \times (\text{length} + \text{breadth}) \]
Given:
- Length (\( l \)) = 49 meters
- New Breadth (\( b \)) = 19 meters
Substitute the given values into the formula:
\[ \text{Perimeter} = 2 \times (49 + 19) \]
\[ \text{Perimeter} = 2 \times 68 \]
\[ \text{Perimeter} = 136 \]
### Answer: 136 meters
Expected answer: 136
Correct? Yes
Problem: A rectangle has a length of 16m and a width of 7m. What is the area of this rectangle?
GPT-4o:
To find the area of a rectangle, we use the formula:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Given:
- Length = 16 meters
- Width = 7 meters
Substitute the given values into the formula:
\[ \text{Area} = 16 \, \text{m} \times 7 \, \text{m} \]
\[ \text{Area} = 112 \, \text{m}^2 \]
### Answer: 112 m²
Expected answer: 112
Correct? Yes
Problem: The perimeter of a rectangular playground is found by the formula 2*(Length + Breadth). If the length of the playground is 14m and the breadth is 2m, what is the perimeter of the playground?
GPT-4o:
To find the perimeter of the rectangular playground, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Breadth}) \]
Given:
- Length (\(L\)) = 14 meters
- Breadth (\(B\)) = 2 meters
First, we calculate the sum of the length and the breadth:
\[ L + B = 14 + 2 = 16 \]
Next, we multiply this sum by 2 to find the perimeter:
\[ \text{Perimeter} = 2 \times 16 = 32 \]
### Answer: 32 meters
Expected answer: 32
Correct? Yes
Problem: Now, consider that the breadth of the rectangular playground is changed to 3m, keeping the length the same as before (14m). What is the new perimeter of the playground?
GPT-4o:
To find the new perimeter of the rectangular playground with the updated breadth, we use the same formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Breadth}) \]
Given:
- Length (\(L\)) = 14 meters
- New Breadth (\(B\)) = 3 meters
First, we calculate the sum of the length and the new breadth:
\[ L + B = 14 + 3 = 17 \]
Next, we multiply this sum by 2 to find the new perimeter:
\[ \text{Perimeter} = 2 \times 17 = 34 \]
### Answer: 34 meters
Expected answer: 34
Correct? Yes
Problem: A square has a side length of 16 units. What is the perimeter of the square?
GPT-4o:
To find the perimeter of a square, we use the formula:
\[ \text{Perimeter} = 4 \times \text{side length} \]
Given that the side length of the square is 16 units, we can substitute this value into the formula:
\[ \text{Perimeter} = 4 \times 16 \]
Calculating this, we get:
\[ \text{Perimeter} = 64 \]
### Answer: 64 units
Expected answer: 64
Correct? Yes
Problem: In the previous problem, the square's side length was 16 units. Now consider a rectangle with a length of 22 units and a width of 16 units. What is the perimeter of this rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{length} + \text{width}) \]
Given that the length of the rectangle is 22 units and the width is 16 units, we can substitute these values into the formula:
\[ \text{Perimeter} = 2 \times (22 + 16) \]
First, we calculate the sum inside the parentheses:
\[ 22 + 16 = 38 \]
Next, we multiply by 2:
\[ \text{Perimeter} = 2 \times 38 = 76 \]
### Answer: 76 units
Expected answer: 76
Correct? Yes
Problem: A rectangular garden has a width of 7 meters and a length of 21 meters. What is the perimeter of this garden?
GPT-4o:
To find the perimeter of a rectangular garden, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Width = 7 meters
- Length = 21 meters
First, we add the length and the width:
\[ \text{Length} + \text{Width} = 21 + 7 = 28 \text{ meters} \]
Next, we multiply the sum by 2 to find the perimeter:
\[ \text{Perimeter} = 2 \times 28 = 56 \text{ meters} \]
### Answer: 56 meters
Expected answer: 56
Correct? Yes
Problem: The width of the garden remains the same, but the length is reduced to 17 meters. Now, what is the new perimeter of the garden?
GPT-4o:
To find the new perimeter of the rectangular garden with the updated length, we use the same perimeter formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Width = 7 meters (remains the same)
- New Length = 17 meters
First, we add the new length and the width:
\[ \text{Length} + \text{Width} = 17 + 7 = 24 \text{ meters} \]
Next, we multiply the sum by 2 to find the new perimeter:
\[ \text{Perimeter} = 2 \times 24 = 48 \text{ meters} \]
### Answer: 48 meters
Expected answer: 48
Correct? Yes
Problem: A rectangle has a length of 20 meters and a width of 5 meters. What is the area of the rectangle?
GPT-4o:
To find the area of a rectangle, we use the formula:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Given:
- Length = 20 meters
- Width = 5 meters
Substituting the given values into the formula:
\[ \text{Area} = 20 \, \text{meters} \times 5 \, \text{meters} \]
\[ \text{Area} = 100 \, \text{square meters} \]
### Answer: 100 square meters
Expected answer: 100
Correct? Yes
Problem: In the previous problem, we calculated the area of a rectangle with a length of 20 meters and a width of 5 meters. This time, instead of having the width as 5 meters, we have a new rectangle where the width is 17 meters. With the length still being 20 meters, what is the area of this new rectangle?
GPT-4o:
To find the area of the new rectangle, we use the same formula:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Given:
- Length = 20 meters
- Width = 17 meters
Substituting the given values into the formula:
\[ \text{Area} = 20 \, \text{meters} \times 17 \, \text{meters} \]
\[ \text{Area} = 340 \, \text{square meters} \]
### Answer: 340 square meters
Expected answer: 340
Correct? Yes
Problem: A rectangle has a length of 37m and a width of 43m. What is the perimeter of this rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 37 meters
- Width (\(W\)) = 43 meters
First, we add the length and the width:
\[ L + W = 37 + 43 = 80 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ \text{Perimeter} = 2 \times 80 = 160 \]
### Answer: 160 meters
Expected answer: 160
Correct? Yes
Problem: Suppose the length of the rectangle is now changed to 32m while the width remains the same at 43m. What will be the new perimeter of the rectangle?
GPT-4o:
To find the new perimeter of the rectangle with the updated length, we use the same perimeter formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- New Length (\(L\)) = 32 meters
- Width (\(W\)) = 43 meters
First, we add the new length and the width:
\[ L + W = 32 + 43 = 75 \]
Next, we multiply the sum by 2 to find the new perimeter:
\[ \text{Perimeter} = 2 \times 75 = 150 \]
### Answer: 150 meters
Expected answer: 150
Correct? Yes
Problem: A rectangle has a length of 9m. If the width of the rectangle is 1m, what is the area?
GPT-4o:
To find the area of a rectangle, we use the formula:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Given:
- Length = 9 meters
- Width = 1 meter
Substitute the given values into the formula:
\[ \text{Area} = 9 \, \text{m} \times 1 \, \text{m} \]
\[ \text{Area} = 9 \, \text{m}^2 \]
### Answer: 9 m²
Expected answer: 9
Correct? Yes
Problem: In the previous problem, we considered a rectangle with a length of 9m and a width of 1m. Now, consider if the width of the rectangle was actually 40m instead of 1m. What would be the new area of the rectangle?
GPT-4o:
To find the area of a rectangle, we use the formula:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Given:
- Length = 9 meters
- Width = 40 meters
Substitute the given values into the formula:
\[ \text{Area} = 9 \, \text{m} \times 40 \, \text{m} \]
\[ \text{Area} = 360 \, \text{m}^2 \]
### Answer: 360 m²
Expected answer: 360
Correct? Yes
Problem: A rectangle's area is the product of 22m and 23m. What is the area of the rectangle?
GPT-4o:
To find the area of the rectangle, we need to multiply the length and the width. Given that the length is 22 meters and the width is 23 meters, we calculate the area as follows:
\[ \text{Area} = \text{Length} \times \text{Width} \]
\[ \text{Area} = 22 \, \text{m} \times 23 \, \text{m} \]
Now, we perform the multiplication:
\[ 22 \times 23 = 506 \]
Therefore, the area of the rectangle is 506 square meters.
### Answer: 506 square meters.
Expected answer: 506
Correct? Yes
Problem: What if we reconsider the area of the rectangle, but this time its width is 6m instead of 23m, while maintaining the length at 22m. What would then be the area of the rectangle?
GPT-4o:
To find the area of the rectangle with the new dimensions, we need to multiply the length by the new width. Given that the length is 22 meters and the new width is 6 meters, we calculate the area as follows:
\[ \text{Area} = \text{Length} \times \text{Width} \]
\[ \text{Area} = 22 \, \text{m} \times 6 \, \text{m} \]
Now, we perform the multiplication:
\[ 22 \times 6 = 132 \]
Therefore, the area of the rectangle with the new width is 132 square meters.
### Answer: 132 square meters.
Expected answer: 132
Correct? Yes
Problem: A rectangle has one side that measures 14 cm and another side that measures 21 cm. What is the perimeter of this rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{length} + \text{width}) \]
Given:
- Length = 21 cm
- Width = 14 cm
First, we add the length and the width:
\[ 21 \, \text{cm} + 14 \, \text{cm} = 35 \, \text{cm} \]
Next, we multiply the sum by 2 to find the perimeter:
\[ 2 \times 35 \, \text{cm} = 70 \, \text{cm} \]
### Answer: 70 cm
Expected answer: 70
Correct? Yes
Problem: Consider the previous rectangle and suppose its side that previously measured 21 cm now measures 47 cm. What is the new perimeter of this rectangle?
GPT-4o:
To find the new perimeter of the rectangle, we use the same formula:
\[ \text{Perimeter} = 2 \times (\text{length} + \text{width}) \]
Given the new dimensions:
- Length = 47 cm
- Width = 14 cm
First, we add the new length and the width:
\[ 47 \, \text{cm} + 14 \, \text{cm} = 61 \, \text{cm} \]
Next, we multiply the sum by 2 to find the new perimeter:
\[ 2 \times 61 \, \text{cm} = 122 \, \text{cm} \]
### Answer: 122 cm
Expected answer: 122
Correct? Yes
Problem: In Clara's backyard, there is a rectangular swimming pool. The length of the pool is 23 meters, while the width is 12 meters. What is the area of the swimming pool in Clara's backyard?
GPT-4o:
To find the area of a rectangular swimming pool, we use the formula for the area of a rectangle:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Given:
- Length = 23 meters
- Width = 12 meters
Substitute the given values into the formula:
\[ \text{Area} = 23 \, \text{meters} \times 12 \, \text{meters} \]
Calculate the product:
\[ \text{Area} = 276 \, \text{square meters} \]
### Answer: 276 square meters
Expected answer: 276
Correct? Yes
Problem: A rectangle has a length of 39 meters and a width of 8 meters. What is the perimeter of this rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length = 39 meters
- Width = 8 meters
First, we add the length and the width:
\[ 39 + 8 = 47 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ 2 \times 47 = 94 \]
### Answer: 94 meters
Expected answer: 94
Correct? Yes
Problem: Suppose the width of the rectangle was actually 23 meters instead of 8 meters. What would the perimeter of the rectangle be in this case?
GPT-4o:
To find the perimeter of the rectangle with the updated width, we use the same formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length = 39 meters
- Width = 23 meters
First, we add the length and the width:
\[ 39 + 23 = 62 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ 2 \times 62 = 124 \]
### Answer: 124 meters
Expected answer: 124
Correct? Yes
Problem: A rectangle has a width of 27m and a length of 40m. What is the perimeter of the rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Width = 27 meters
- Length = 40 meters
First, we add the length and the width:
\[ 40 + 27 = 67 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ 2 \times 67 = 134 \]
### Answer: 134 meters
Expected answer: 134
Correct? Yes
Problem: What if the length of the rectangle was 44m instead of 40m while keeping the width still at 27m? What would the new perimeter of the rectangle be?
GPT-4o:
To find the new perimeter of the rectangle with the updated length, we use the same formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Width = 27 meters
- New Length = 44 meters
First, we add the new length and the width:
\[ 44 + 27 = 71 \]
Next, we multiply the sum by 2 to find the new perimeter:
\[ 2 \times 71 = 142 \]
### Answer: 142 meters
Expected answer: 142
Correct? Yes
Problem: A rectangle's length is 35m and its width is 9m. What is the perimeter of the rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 35 meters
- Width (\(W\)) = 9 meters
First, we add the length and the width:
\[ L + W = 35 + 9 = 44 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ \text{Perimeter} = 2 \times 44 = 88 \]
### Answer: 88 meters
Expected answer: 88
Correct? Yes
Problem: Suppose we resized the rectangle and shortened its length from 35m to 19m, while keeping the width the same. What would the new perimeter of the rectangle be?
GPT-4o:
To find the new perimeter of the resized rectangle, we use the same perimeter formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given the new dimensions:
- New Length (\(L\)) = 19 meters
- Width (\(W\)) = 9 meters
First, we add the new length and the width:
\[ L + W = 19 + 9 = 28 \]
Next, we multiply the sum by 2 to find the new perimeter:
\[ \text{Perimeter} = 2 \times 28 = 56 \]
### Answer: 56 meters
Expected answer: 56
Correct? Yes
Problem: A rectangle has a length of 38m and a width of 11m. What is the perimeter of this rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\( L \)) = 38 meters
- Width (\( W \)) = 11 meters
First, we add the length and the width:
\[ L + W = 38 + 11 = 49 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ \text{Perimeter} = 2 \times 49 = 98 \]
### Answer: 98 meters
Expected answer: 98
Correct? Yes
Problem: In the rectangle mentioned earlier, suppose we now construct another rectangle such that its length remains the same (38m), but the width is now 23m instead of 11m. How would the perimeter of this new rectangle be calculated?
GPT-4o:
To find the perimeter of the new rectangle, we use the same formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given for the new rectangle:
- Length (\( L \)) = 38 meters
- Width (\( W \)) = 23 meters
First, we add the length and the width:
\[ L + W = 38 + 23 = 61 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ \text{Perimeter} = 2 \times 61 = 122 \]
### Answer: 122 meters
Expected answer: 122
Correct? Yes
Problem: One side of a rectangle is 28m long, and the other side is 29m long. What is the area of this rectangle?
GPT-4o:
To find the area of a rectangle, we use the formula:
\[ \text{Area} = \text{length} \times \text{width} \]
Given:
- Length = 28 meters
- Width = 29 meters
Substitute the given values into the formula:
\[ \text{Area} = 28 \, \text{m} \times 29 \, \text{m} \]
Now, perform the multiplication:
\[ \text{Area} = 812 \, \text{m}^2 \]
### Answer: 812 m²
Expected answer: 812
Correct? Yes
Problem: Suppose the side of the rectangle previously mentioned gets extended to 35m instead of 29m, while the other side remains 28m. What would be the new area of the rectangle in this case?
GPT-4o:
To find the new area of the rectangle after extending one side to 35 meters while the other side remains 28 meters, we use the same area formula:
\[ \text{Area} = \text{length} \times \text{width} \]
Given:
- Length = 28 meters
- New Width = 35 meters
Substitute the given values into the formula:
\[ \text{Area} = 28 \, \text{m} \times 35 \, \text{m} \]
Now, perform the multiplication:
\[ \text{Area} = 980 \, \text{m}^2 \]
### Answer: 980 m²
Expected answer: 980
Correct? Yes
Problem: A square has a side length of 9m. What is the total length of fencing needed to enclose the square's perimeter?
GPT-4o:
To find the total length of fencing needed to enclose the square's perimeter, we need to calculate the perimeter of the square. The formula for the perimeter \( P \) of a square with side length \( s \) is: \[ P = 4s \] Given that the side length \( s \) is 9 meters, we can substitute this value into the formula: \[ P = 4 \times 9 \] \[ P = 36 \] Therefore, the total length of fencing needed is 36 meters. ### Answer: 36 meters
Expected answer: 36
Correct? Yes
Problem: A rectangle has a length of 38m and a width of 29m. What is the perimeter of this rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 38 meters
- Width (\(W\)) = 29 meters
First, we add the length and the width:
\[ L + W = 38 + 29 = 67 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ \text{Perimeter} = 2 \times 67 = 134 \]
### Answer: 134 meters
Expected answer: 134
Correct? Yes
Problem: Now consider that the rectangle from the previous question actually had a length of 6m instead of 38m. What would be the new perimeter of the rectangle?
GPT-4o:
To find the new perimeter of the rectangle with the updated length, we use the same formula for the perimeter of a rectangle:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- New Length (\(L\)) = 6 meters
- Width (\(W\)) = 29 meters
First, we add the new length and the width:
\[ L + W = 6 + 29 = 35 \]
Next, we multiply the sum by 2 to find the new perimeter:
\[ \text{Perimeter} = 2 \times 35 = 70 \]
### Answer: 70 meters
Expected answer: 70
Correct? Yes
Problem: A rectangle has one side of length 45m and another side of length 47m. Find the perimeter of the rectangle.
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{length} + \text{width}) \]
Given:
- Length = 45 meters
- Width = 47 meters
First, we add the length and the width:
\[ 45 + 47 = 92 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ 2 \times 92 = 184 \]
### Answer: 184 meters
Expected answer: 184
Correct? Yes
Problem: Suppose the length of one side of the rectangle is altered from 47m to 46m. What would be the new perimeter of the rectangle now?
GPT-4o:
To find the new perimeter of the rectangle after altering one side, we use the same perimeter formula:
\[ \text{Perimeter} = 2 \times (\text{length} + \text{width}) \]
Given:
- Length = 45 meters
- New width = 46 meters
First, we add the new length and width:
\[ 45 + 46 = 91 \]
Next, we multiply the sum by 2 to find the new perimeter:
\[ 2 \times 91 = 182 \]
### Answer: 182 meters
Expected answer: 182
Correct? Yes
Problem: Given a rectangle where the length is 24m and the width is 20m, what is the perimeter of this rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 24 meters
- Width (\(W\)) = 20 meters
First, we add the length and the width:
\[ L + W = 24 + 20 = 44 \]
Next, we multiply the sum by 2 to find the perimeter:
\[ \text{Perimeter} = 2 \times 44 = 88 \]
### Answer: 88 meters
Expected answer: 88
Correct? Yes
Problem: Continuing from the previous scenario, imagine the length of the rectangle stayed the same at 24m, but the width was reduced to 17m. Now, what would be the new perimeter of this rectangle?
GPT-4o:
To find the new perimeter of the rectangle with the updated width, we use the same perimeter formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length (\(L\)) = 24 meters
- New Width (\(W\)) = 17 meters
First, we add the length and the new width:
\[ L + W = 24 + 17 = 41 \]
Next, we multiply the sum by 2 to find the new perimeter:
\[ \text{Perimeter} = 2 \times 41 = 82 \]
### Answer: 82 meters
Expected answer: 82
Correct? Yes
Problem: The length and width of a rectangle are both 32m. What is the perimeter of this rectangle?
GPT-4o:
To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given that both the length and the width of the rectangle are 32 meters, we can substitute these values into the formula:
\[ \text{Perimeter} = 2 \times (32 \, \text{m} + 32 \, \text{m}) \]
First, add the length and the width:
\[ 32 \, \text{m} + 32 \, \text{m} = 64 \, \text{m} \]
Next, multiply by 2:
\[ 2 \times 64 \, \text{m} = 128 \, \text{m} \]
### Answer: 128 m
Expected answer: 128
Correct? Yes
Problem: The length of the rectangle from the previous problem remains the same, but the width of the rectangle is now adjusted to 13m. What is the new perimeter of this rectangle?
GPT-4o:
To find the new perimeter of the rectangle with the updated width, we use the same perimeter formula:
\[ \text{Perimeter} = 2 \times (\text{Length} + \text{Width}) \]
Given:
- Length = 32 meters
- Width = 13 meters
Substitute these values into the formula:
\[ \text{Perimeter} = 2 \times (32 \, \text{m} + 13 \, \text{m}) \]
First, add the length and the width:
\[ 32 \, \text{m} + 13 \, \text{m} = 45 \, \text{m} \]
Next, multiply by 2:
\[ 2 \times 45 \, \text{m} = 90 \, \text{m} \]
### Answer: 90 m
Expected answer: 90
Correct? Yes