Claude-3 Sonnet on 8.EE.A.2
Common Core Standard 8.EE.A.2
This standard covers the following:
Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that the square root of 2 is irrational.
Problem: A child builds a sandcastle in a sandbox. The sandbox has the shape of a cubic box with 512 cubic feet of sand. What is the length of each side of the sandbox in feet?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of a cube given its volume. Given information: - The sandbox has the shape of a cubic box. - The volume of the sandbox is 512 cubic feet. Let's denote the side length of the cube as x. Since the volume of a cube is given by the formula: Volume = side length^3 We can write: 512 = x^3 Taking the cube root of both sides: x = ∛512 x ≈ 8 feet Therefore, the length of each side of the sandbox is approximately 8 feet. ### Answer: 8
Expected answer: 8
Correct? Yes
Problem: An artisan carves a cubic fireplace out of stone, which has a volume of 729 cubic feet. What is the length of each side of the fireplace in feet?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of the cube given its volume. Given information: - The fireplace is a cube. - The volume of the cube is 729 cubic feet. Step 1: Find the side length of the cube using the formula for the volume of a cube. Volume of a cube = side length × side length × side length 729 cubic feet = side length^3 Step 2: Take the cube root of 729 to find the side length. side length = ∛729 side length = 9 feet Therefore, the length of each side of the cubic fireplace is 9 feet. ### Answer: 9
Expected answer: 9
Correct? Yes
Problem: A farmer has a large cubical barn with a volume of 1000 cubic meters. What is the length of each side of the barn in meters?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of a cube given its volume. Given information: - The barn is a cube. - The volume of the barn is 1000 cubic meters. The formula for the volume of a cube is: Volume = side length × side length × side length or Volume = (side length)^3 Let's represent the side length as s. 1000 = s^3 Taking the cube root of both sides: s = ∛1000 s ≈ 10 meters Therefore, the length of each side of the barn is approximately 10 meters. ### Answer: 10
Expected answer: 10
Correct? Yes
Problem: A giraffe's tank in the zoo is in a cubic shape in order to facilitate its movement. The volume of the tank is 512 m^3. If all sides of the tank are equal in size, what is the size of one side of the tank in meters?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of the cubic tank given its volume. Given information: - The tank is a cube. - The volume of the tank is 512 m^3. Step 1: Find the side length of the cube using the formula for the volume of a cube. Volume of a cube = side length^3 512 m^3 = side length^3 side length = ∛512 side length = 8 meters Therefore, the size of one side of the cubic tank is 8 meters. ### Answer: 8
Expected answer: 8
Correct? Yes
Problem: A mountain has a cuboidal shape and its volume is 64 km^3. If all of the sides of this cuboid are of equal length, what is the length of each side in kilometers?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of the cube given its volume. Given information: - The mountain has a cuboidal (cube) shape. - The volume of the cube is 64 km^3. Step 1: Find the side length of the cube using the formula for the volume of a cube. Volume of a cube = side length × side length × side length 64 km^3 = side length^3 Step 2: Take the cube root of both sides to find the side length. side length = ∛(64 km^3) side length = 4 km Therefore, the length of each side of the cuboidal mountain is 4 km. ### Answer: 4
Expected answer: 4
Correct? Yes
Problem: There's a large crater on the moon that has a cubic shape and a volume of 512 cubic kilometers. What is the length of each edge of this crater in kilometers?
Claude-3 Sonnet:
To solve this problem, we need to find the length of each edge of the cubic crater given its volume. Given information: - The crater has a cubic shape. - The volume of the crater is 512 cubic kilometers. Step 1: Find the formula for the volume of a cube. Volume of a cube = Edge length × Edge length × Edge length Volume of a cube = (Edge length)^3 Step 2: Substitute the given volume and solve for the edge length. 512 = (Edge length)^3 Edge length = ∛512 Edge length = 8 kilometers Therefore, the length of each edge of the cubic crater on the moon is 8 kilometers. ### Answer: 8
Expected answer: 8
Correct? Yes
Problem: A baby is playing with blocks, assembling them to build a larger cube structure. If the total number of blocks is 343, what is the length (represented in blocks) of one side of the cube?
Claude-3 Sonnet:
To solve this problem, we need to find the cube root of 343, as the total number of blocks is a perfect cube. Given information: - The total number of blocks is 343. - The blocks are assembled to build a larger cube structure. Step 1: Find the cube root of 343. Cube root of 343 = 7 Step 2: The length of one side of the cube is equal to the cube root of the total number of blocks. Length of one side of the cube = 7 blocks Therefore, the length of one side of the cube structure made up of 343 blocks is 7 blocks. ### Answer: 7
Expected answer: 7
Correct? Yes
Problem: An alien in a UFO spies a cubic container with a mysterious substance on earth. The volume of the substance is 512 cubic feet. The alien wonders, what is the length of one side of the cubic container in feet?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of a cube given its volume. Given information: - The volume of the substance in the cubic container is 512 cubic feet. The formula for the volume of a cube is: Volume = side length × side length × side length or Volume = (side length)^3 We can substitute the given volume and solve for the side length. 512 cubic feet = (side length)^3 Taking the cube root of both sides: side length = ∛512 side length = 8 feet Therefore, the length of one side of the cubic container is 8 feet. ### Answer: 8
Expected answer: 8
Correct? Yes
Problem: A pencil manufacturer makes pencil-eraser cubes and the volume of each cube is found to be 512 mm^3. What is the edge length of each eraser cube?
Claude-3 Sonnet:
To solve this problem, we need to find the edge length of a cube given its volume. Given information: - The volume of each eraser cube is 512 mm^3. The formula for the volume of a cube is: Volume = edge length × edge length × edge length or Volume = (edge length)^3 We can substitute the given volume and solve for the edge length. 512 mm^3 = (edge length)^3 edge length = ∛(512 mm^3) edge length = 8 mm Therefore, the edge length of each eraser cube is 8 mm. ### Answer: 8 mm
Expected answer: 8
Correct? Yes
Problem: The letter G in a certain font uses a cube as a model for its design. If the volume of this cubic model is 8 cubic cm, what is the length of each side of the cube in cm?
Claude-3 Sonnet:
To solve this problem, we need to use the formula for the volume of a cube, which is given by: Volume = side length × side length × side length or V = s^3 Given information: - The volume of the cubic model is 8 cubic cm. Let's denote the side length of the cube as s. Step 1: Set up the equation using the given information. V = s^3 8 = s^3 Step 2: Take the cube root of both sides to find the side length. s = ∛8 s = 2 cm Therefore, the length of each side of the cube is 2 cm. ### Answer: 2
Expected answer: 2
Correct? Yes
Problem: A shark's territory within the ocean is allotted in the shape of a cube with a volume of 27 cubic kilometers. Can you calculate the length in kilometers of one edge of this cube?
Claude-3 Sonnet:
To solve this problem, we need to find the length of one edge of the cube given its volume. Given information: - The shark's territory is in the shape of a cube. - The volume of the cube is 27 cubic kilometers. The formula for the volume of a cube is: Volume = edge length × edge length × edge length or Volume = (edge length)^3 We can substitute the given volume and solve for the edge length. 27 = (edge length)^3 edge length = ∛27 edge length = 3 kilometers Therefore, the length of one edge of the cube representing the shark's territory is 3 kilometers. ### Answer: 3
Expected answer: 3
Correct? Yes
Problem: A deck of cards is designed in a cubic box. The volume of the box is 1 cubic inch. What is the length of each side of the box in inches?
Claude-3 Sonnet:
To solve this problem, we need to find the length of each side of the cube given its volume. Given information: - The volume of the cubic box is 1 cubic inch. Step 1: Write the formula for the volume of a cube. Volume of a cube = side length × side length × side length Volume of a cube = (side length)^3 Step 2: Substitute the given volume and solve for the side length. 1 cubic inch = (side length)^3 side length = ∛(1 cubic inch) side length = ∛1 side length = 1 inch Therefore, the length of each side of the cubic box is 1 inch. ### Answer: 1
Expected answer: 1
Correct? Yes
Problem: A unique type of flower grows in the shape of a cube. Each cubic centimeter of this flower holds 1 milliliter of nectar. If a particular flower holds 125 milliliters of nectar, what is the length of one side of this cubic flower in centimeters?
Claude-3 Sonnet:
To solve this problem, we need to find the volume of the cubic flower in cubic centimeters and then take the cube root of that volume to find the length of one side. Given information: - Each cubic centimeter of the flower holds 1 milliliter of nectar. - The flower holds 125 milliliters of nectar. Step 1: Find the volume of the flower in cubic centimeters. Volume of the flower in cubic centimeters = Volume of the flower in milliliters Volume of the flower in cubic centimeters = 125 cubic centimeters Step 2: Find the length of one side of the cubic flower. Let the length of one side be x centimeters. Volume of a cube = x^3 cubic centimeters x^3 = 125 x = ∛125 x = 5 centimeters Therefore, the length of one side of the cubic flower is 5 centimeters. ### Answer: 5
Expected answer: 5
Correct? Yes
Problem: In a forest, the volume of a cubical tree stump was found to be 125 cubic meters. What is the length of one side of the tree stump?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of the cubical tree stump given its volume. Given information: - The tree stump is a cube. - The volume of the tree stump is 125 cubic meters. Step 1: Recall the formula for the volume of a cube. Volume of a cube = side length × side length × side length Volume = s^3 Step 2: Substitute the given volume and solve for the side length (s). 125 = s^3 s = ∛125 s = 5 meters Therefore, the length of one side of the cubical tree stump is 5 meters. ### Answer: 5
Expected answer: 5
Correct? Yes
Problem: A jeweler is designing a bracelet in the shape of a cube. The volume of the bracelet is 1000 cubic millimeters. What is the length of each edge of the bracelet in millimeters?
Claude-3 Sonnet:
To solve this problem, we need to find the edge length of a cube given its volume. Given information: - The volume of the bracelet (cube) is 1000 cubic millimeters. Step 1: Write the formula for the volume of a cube. Volume of a cube = Edge length × Edge length × Edge length Volume of a cube = (Edge length)^3 Step 2: Substitute the given volume and solve for the edge length. 1000 = (Edge length)^3 Edge length = ∛1000 Edge length = 10 millimeters Therefore, the length of each edge of the bracelet is 10 millimeters. ### Answer: 10
Expected answer: 10
Correct? Yes
Problem: A bar of soap is in the shape of a cube. If its volume is 1 cubic cm, what is the length of each side of the soap in cm?
Claude-3 Sonnet:
To solve this problem, we need to use the formula for the volume of a cube, which is given by: Volume = side length × side length × side length or V = s^3 Given information: - The volume of the soap cube is 1 cubic cm. Let's denote the side length of the cube as s. Step 1: Substitute the given volume and the side length variable into the volume formula. 1 cm^3 = s^3 Step 2: Take the cube root of both sides to find the side length. s = ∛(1 cm^3) s = 1 cm Therefore, the length of each side of the soap cube is 1 cm. ### Answer: 1 cm
Expected answer: 1
Correct? Yes
Problem: A forest ranger discovered a rare cubic yew tree with a volume of 216 cubic meters. What is the length of each edge of this yew tree in meters?
Claude-3 Sonnet:
To solve this problem, we need to find the edge length of a cube given its volume. Given information: - The volume of the cubic yew tree is 216 cubic meters. The formula for the volume of a cube is: Volume = edge length × edge length × edge length or Volume = (edge length)^3 We can rearrange the formula to find the edge length: edge length = ∛(Volume) Substituting the given volume, we get: edge length = ∛(216) edge length = 6 meters Therefore, the length of each edge of the cubic yew tree is 6 meters. ### Answer: 6
Expected answer: 6
Correct? Yes
Problem: A blanket is folded in a perfect cube shape and has a volume of 1000 cubic cm. What is the length of each side of the blanket in centimeters?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of the cube given its volume. Given information: - The blanket is folded into a perfect cube shape. - The volume of the cube is 1000 cubic cm. Step 1: Find the formula for the volume of a cube. Volume of a cube = side length × side length × side length Volume of a cube = (side length)^3 Step 2: Substitute the given volume and solve for the side length. 1000 cubic cm = (side length)^3 side length = ∛(1000 cubic cm) side length = 10 cm Therefore, the length of each side of the blanket when folded into a perfect cube is 10 cm. ### Answer: 10
Expected answer: 10
Correct? Yes
Problem: A toy train is positioned inside a box, which is a cube. The volume of the box is 125 cubic centimeters. What is the length of each side of the box in centimeters?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of the cube given its volume. Given information: - The box is a cube. - The volume of the cube is 125 cubic centimeters. Step 1: Find the formula for the volume of a cube. Volume of a cube = side length × side length × side length Volume of a cube = (side length)^3 Step 2: Substitute the given volume and solve for the side length. 125 = (side length)^3 side length = ∛125 side length = 5 centimeters Therefore, the length of each side of the cube (box) is 5 centimeters. ### Answer: 5
Expected answer: 5
Correct? Yes
Problem: A perfume bottle is shaped like a cube and it has a volume of 125 cubic centimeters. What is the length of each side of the bottle in centimeters?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of a cube given its volume. Given information: - The perfume bottle is shaped like a cube. - The volume of the cube is 125 cubic centimeters. Step 1: Write the formula for the volume of a cube. Volume of a cube = side length × side length × side length Volume of a cube = (side length)^3 Step 2: Substitute the given volume and solve for the side length. 125 = (side length)^3 side length = ∛125 side length = 5 cm Therefore, the length of each side of the perfume bottle is 5 centimeters. ### Answer: 5
Expected answer: 5
Correct? Yes
Problem: A pencil-shaped building has a cuboidal structure. The building has a uniform cross-sectional area all along its height. When the volume of the building is calculated, it comes out to be 216 cubic meters. If the building height, width and depth are all equal, what is the length in meters of one side?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of the cube, given that the volume is 216 cubic meters. Given information: - The building has a cuboidal structure with a uniform cross-sectional area along its height. - The volume of the building is 216 cubic meters. - The height, width, and depth of the building are all equal. Let's denote the side length of the cube as x. Since the volume of a cube is given by the formula: Volume = side length^3 We have: 216 = x^3 Taking the cube root of both sides: x = ∛216 Using a calculator or mathematical tables, we find that: x = 6 meters Therefore, the length of one side of the building is 6 meters. ### Answer: 6
Expected answer: 6
Correct? Yes
Problem: A chair is designed in the shape of a cube, and its volume is 343 cubic feet. What is the length of each side of the chair in feet?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of the cube given its volume. Given information: - The chair is designed in the shape of a cube. - The volume of the cube is 343 cubic feet. Step 1: Find the side length of the cube using the formula for the volume of a cube. Volume of a cube = side length × side length × side length 343 cubic feet = side length^3 Step 2: Take the cube root of 343 to find the side length. side length = ∛343 side length = 7 feet Therefore, the length of each side of the chair is 7 feet. ### Answer: 7
Expected answer: 7
Correct? Yes
Problem: A roller skate company designs a new line of cubes that will go on the end of each skate. The volume of each cube is calculated to be 729 cubic inches. What is the length, in inches, of a side of each cube?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of a cube given its volume. Given information: - The volume of each cube is 729 cubic inches. The formula for the volume of a cube is: Volume = side length × side length × side length or Volume = (side length)^3 We can set up an equation: 729 = (side length)^3 To find the side length, we need to take the cube root of 729. side length = ∛729 side length = 9 inches Therefore, the length of a side of each cube is 9 inches. ### Answer: 9
Expected answer: 9
Correct? Yes
Problem: An alien species uses cubical spaceships for traveling. The volume of one of these spaceships is 8 cubic miles. What is the length in miles of each edge of the spaceship?
Claude-3 Sonnet:
To solve this problem, we need to find the edge length of a cube given its volume. Given information: - The volume of the spaceship is 8 cubic miles. - The spaceship is a cube. The formula for the volume of a cube is: Volume = edge length × edge length × edge length or Volume = (edge length)^3 Let's represent the edge length as x. 8 = x^3 (substituting the given volume) x = ∛8 (taking the cube root of both sides) x ≈ 2 miles (calculating the cube root) Therefore, the length of each edge of the spaceship is approximately 2 miles. ### Answer: 2
Expected answer: 2
Correct? Yes
Problem: A strawberry farmer builds a cube-shaped cold storage unit for his strawberries. The cold storage has a volume of 343 cubic feet. What is the length of each side of the storage unit in feet?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of a cube given its volume. Given information: - The cold storage unit is a cube. - The volume of the cold storage unit is 343 cubic feet. Step 1: Find the side length of the cube using the formula for the volume of a cube. Volume of a cube = side length × side length × side length 343 cubic feet = side length^3 Step 2: Take the cube root of 343 to find the side length. side length = ∛343 side length = 7 feet Therefore, the length of each side of the cold storage unit is 7 feet. ### Answer: 7
Expected answer: 7
Correct? Yes
Problem: The volume of the compass cube is 8 cubic inches. What is the length of the edge of the compass cube in inches?
Claude-3 Sonnet:
To solve this problem, we need to find the edge length of the cube given its volume. Given information: - The volume of the cube is 8 cubic inches. The formula for the volume of a cube is: Volume = edge length × edge length × edge length or Volume = (edge length)^3 Let's represent the edge length as x. We know that the volume is 8 cubic inches, so we can substitute the values in the formula: 8 = x^3 Taking the cube root of both sides: x = ∛8 Using a calculator or looking up the cube root of 8, we get: x = 2 inches Therefore, the length of the edge of the compass cube is 2 inches. ### Answer: 2 inches
Expected answer: 2
Correct? Yes
Problem: A potato farmer has a unique farm and prefers to grow his potatoes in cubic plots. One plot has a volume that measures 729 cubic meters. What is the length of each side of the plot in meters?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of a cube given its volume. Given information: - The volume of the cubic plot is 729 cubic meters. The volume of a cube is given by the formula: Volume = side length × side length × side length or Volume = (side length)^3 We want to find the side length, given the volume. Step 1: Take the cube root of the volume to find the side length. side length = ∛(729) side length = 9 meters Therefore, the length of each side of the cubic plot is 9 meters. ### Answer: 9
Expected answer: 9
Correct? Yes
Problem: A slide in the amusement park has a cuboid shape. The volume of this slide is given as 1000 cubic feet. If the slide has the same length, width, and height, what is the length of one side of the slide in feet?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of a cube with a volume of 1000 cubic feet. Given information: - The volume of the slide is 1000 cubic feet. - The slide has the same length, width, and height (it is a cube). Let's denote the side length of the cube as x. The volume of a cube is given by the formula: Volume = side length × side length × side length or V = x^3 We know that the volume is 1000 cubic feet, so we can substitute this value into the formula: 1000 = x^3 To find the value of x, we need to take the cube root of 1000. x = ∛1000 x ≈ 10 feet Therefore, the length of one side of the slide is approximately 10 feet. ### Answer: 10
Expected answer: 10
Correct? Yes
Problem: A zookeeper is building a new play cube for the monkeys. The volume of the cube is huge - exactly 512 cubic feet! The zookeeper wants all sides of the play cube to be of equal length. What is the length of each side of the play cube in feet?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of a cube given its volume. Given information: - The volume of the play cube is 512 cubic feet. - The cube has equal side lengths. The formula for the volume of a cube is: Volume = side length × side length × side length or Volume = (side length)^3 Let's represent the side length as x. We know that the volume is 512 cubic feet, so we can substitute the values in the formula: 512 = x^3 To find the value of x, we need to take the cube root of 512. x = ∛512 x = 8 feet Therefore, the length of each side of the play cube is 8 feet. ### Answer: 8
Expected answer: 8
Correct? Yes
Problem: A magical hat inflates into the shape of a perfect cube. When fully inflated, its volume measures 27 cubic centimeters. Can you figure out the length of one edge of the hat when it is fully inflated?
Claude-3 Sonnet:
To solve this problem, we need to use the formula for the volume of a cube, which is given by: Volume = side length × side length × side length or V = s^3 Given information: - The volume of the fully inflated hat is 27 cubic centimeters. We need to find the side length (s) of the cube. Step 1: Set up the equation using the given volume. 27 = s^3 Step 2: Take the cube root of both sides to find the side length. s = ∛27 s = 3 cm Therefore, when the magical hat is fully inflated into the shape of a perfect cube, the length of one edge is 3 centimeters. ### Answer: 3 cm
Expected answer: 3
Correct? Yes
Problem: A grape farmer is packing his grapes into cube-shaped containers. If one such container holds 27 grapes, how many grapes does each side of the cube hold?
Claude-3 Sonnet:
To solve this problem, we need to find the cube root of 27, as it represents the length of each side of the cube-shaped container. Given information: - The container is cube-shaped. - The container holds 27 grapes. Step 1: Find the cube root of 27. Cube root of 27 = 3 Therefore, each side of the cube-shaped container holds 3 grapes. ### Answer: 3
Expected answer: 3
Correct? Yes
Problem: A truck driver transports cargo in a big cube-shaped container. He knows that the volume of the container is given as 216 cubic meters. Can you help him find out the length of one side of the cube container?
Claude-3 Sonnet:
To find the length of one side of a cube-shaped container, we need to use the formula for the volume of a cube, which is: Volume = side length × side length × side length or V = s^3 Given information: - The volume of the cube-shaped container is 216 cubic meters. Let's represent the side length as s. We can substitute the given volume and the side length variable into the formula: 216 = s^3 To find the value of s, we need to take the cube root of 216. s = ∛216 s ≈ 6 meters Therefore, the length of one side of the cube-shaped container is approximately 6 meters. ### Answer: 6 meters
Expected answer: 6
Correct? Yes
Problem: A cube-shaped bracelet charm has a volume of 64 mm^3. What is the length of each edge of the cube charm in millimeters?
Claude-3 Sonnet:
To solve this problem, we need to find the edge length of the cube given its volume. Given information: - The bracelet charm is a cube. - The volume of the cube is 64 mm^3. The formula for the volume of a cube is: Volume = edge length × edge length × edge length or Volume = (edge length)^3 We can substitute the given volume and solve for the edge length. 64 mm^3 = (edge length)^3 edge length = ∛(64 mm^3) edge length = 4 mm Therefore, the length of each edge of the cube charm is 4 millimeters. ### Answer: 4
Expected answer: 4
Correct? Yes
Problem: If a number is cubed and the result is 64, what is the number?
Claude-3 Sonnet:
To solve this problem, we need to find the number whose cube (third power) is equal to 64. Given information: - The cube of a number is 64. We can represent this as: x^3 = 64 To find the value of x, we need to take the cube root of 64. The cube root of a number is the value that, when raised to the power of 3, gives the original number. The cube root of 64 can be calculated as: ∛64 = 4 Therefore, the number whose cube is 64 is 4. ### Answer: 4
Expected answer: 4
Correct? Yes
Problem: Your petite mug is perfect for hot chocolate and has a cubic shape, accommodating comfy 1 liter of hot drink. Can you tell me, how long, in liters, is each side of the petite mug?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of a cube with a volume of 1 liter. Given information: - The mug has a cubic shape. - The mug can accommodate 1 liter of hot drink. Step 1: Find the volume of a cube in terms of its side length. Volume of a cube = side length × side length × side length Volume of a cube = (side length)^3 Step 2: Substitute the given volume of 1 liter and solve for the side length. 1 liter = (side length)^3 side length = ∛(1 liter) side length = ∛1 side length = 1 Therefore, each side of the petite mug has a length of 1 liter. ### Answer: 1
Expected answer: 1
Correct? Yes
Problem: The volume of a cube-shaped blanket is 343 cubic inches when fully fluffed. What is the length of one side of this fluffed blanket in inches?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of the cube given its volume. Given information: - The volume of the cube-shaped blanket is 343 cubic inches when fully fluffed. The volume of a cube is given by the formula: Volume = side length × side length × side length or Volume = (side length)^3 We can set up an equation: 343 = (side length)^3 Taking the cube root of both sides: side length = ∛343 side length = 7 inches Therefore, the length of one side of the fluffed blanket is 7 inches. ### Answer: 7
Expected answer: 7
Correct? Yes
Problem: A Penguin has stacked ice cubes in a specific way to form a larger perfect cube. The volume of the larger cube made up of smaller cubes is 729 cubic cm. If each side of the smaller ice cube is the same length in cm, what is that length?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of the smaller ice cubes that make up the larger perfect cube. Given information: - The volume of the larger cube made up of smaller cubes is 729 cubic cm. - The larger cube is a perfect cube, meaning all its dimensions are equal. Step 1: Find the side length of the larger cube. Volume of a cube = Side length × Side length × Side length 729 cubic cm = Side length^3 Side length = ∛729 (cube root of 729) Side length = 9 cm Step 2: Find the number of smaller cubes that make up the larger cube. Number of smaller cubes = (Side length of larger cube)^3 Number of smaller cubes = 9^3 = 729 smaller cubes Step 3: Find the side length of each smaller cube. Since the larger cube is made up of smaller cubes, the volume of the larger cube is equal to the sum of the volumes of all the smaller cubes. Volume of larger cube = Sum of volumes of smaller cubes 729 cubic cm = (Side length of smaller cube)^3 × 729 (Side length of smaller cube)^3 = 729 / 729 Side length of smaller cube = ∛1 Side length of smaller cube = 1 cm Therefore, the length of each side of the smaller ice cubes is 1 cm. ### Answer: 1
Expected answer: 9
Correct? No
Problem: A kid on a scooter notices that his new scooter wheel, which has a cubic shape, has a volume of 216 cubic cm. What is the length of each side of the wheel in cm?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of the cube given its volume. Given information: - The scooter wheel has a cubic shape. - The volume of the wheel is 216 cubic cm. The formula for the volume of a cube is: Volume = side length × side length × side length or Volume = (side length)^3 We can substitute the given volume and solve for the side length. 216 cm^3 = (side length)^3 side length = ∛(216 cm^3) side length = 6 cm Therefore, the length of each side of the cubic scooter wheel is 6 cm. ### Answer: 6
Expected answer: 6
Correct? Yes
Problem: A butterfly places an egg in a cubic flower bud. The volume of the bud is 216 cubic millimeters. What length, in millimeters, is each edge of this flower bud?
Claude-3 Sonnet:
To solve this problem, we need to find the edge length of a cube given its volume. Given information: - The flower bud is a cube. - The volume of the flower bud is 216 cubic millimeters. Step 1: Find the formula for the volume of a cube. Volume of a cube = Edge length × Edge length × Edge length Volume of a cube = (Edge length)^3 Step 2: Substitute the given volume and solve for the edge length. 216 = (Edge length)^3 Edge length = ∛216 Edge length = 6 millimeters Therefore, each edge of the flower bud is 6 millimeters long. ### Answer: 6
Expected answer: 6
Correct? Yes
Problem: A snail carries a shell that is shaped like a cube. The volume of the shell is 27 cm^3. What is the length of each side of the snail's cubical shell in centimeters?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of the cube given its volume. Given information: - The shell is shaped like a cube. - The volume of the shell is 27 cm^3. The formula for the volume of a cube is: Volume = side length × side length × side length or Volume = (side length)^3 We know that the volume is 27 cm^3, so we can substitute this value into the formula and solve for the side length. 27 cm^3 = (side length)^3 side length = ∛(27 cm^3) side length = 3 cm Therefore, the length of each side of the snail's cubical shell is 3 centimeters. ### Answer: 3
Expected answer: 3
Correct? Yes
Problem: For a party, a giant cubic party balloon is being used. If the volume of the balloon is 27 cubic feet, what is the length of one side of the balloon in feet?
Claude-3 Sonnet:
To solve this problem, we need to use the formula for the volume of a cube, which is given by: Volume = side length × side length × side length or V = s^3 Given information: - The volume of the cubic party balloon is 27 cubic feet. We need to find the side length (s) of the cube. Step 1: Set up the equation using the given volume. 27 = s^3 Step 2: Take the cube root of both sides to find the side length. s = ∛27 s = 3 feet Therefore, the length of one side of the cubic party balloon is 3 feet. ### Answer: 3
Expected answer: 3
Correct? Yes
Problem: A firefighter is pumping water out of a cubic tank to fight a fire. If the tank has a volume of 216 cubic feet, what is the length of one edge of the tank in feet?
Claude-3 Sonnet:
To solve this problem, we need to find the edge length of a cube given its volume. Given information: - The tank is a cube. - The volume of the tank is 216 cubic feet. Step 1: Find the relationship between the volume and the edge length of a cube. Volume of a cube = Edge length × Edge length × Edge length Volume of a cube = (Edge length)^3 Step 2: Substitute the given volume and solve for the edge length. 216 cubic feet = (Edge length)^3 Edge length = ∛216 Edge length = 6 feet Therefore, the length of one edge of the cubic tank is 6 feet. ### Answer: 6
Expected answer: 6
Correct? Yes
Problem: You have a cube-shaped box of cards with a volume of 216 cubic inches. What is the length of one side of the box in inches?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of the cube given its volume. Given information: - The box is cube-shaped. - The volume of the box is 216 cubic inches. Step 1: Find the formula for the volume of a cube. Volume of a cube = side length × side length × side length Volume of a cube = (side length)^3 Step 2: Substitute the given volume and solve for the side length. 216 cubic inches = (side length)^3 side length = ∛216 side length = 6 inches Therefore, the length of one side of the cube-shaped box is 6 inches. ### Answer: 6
Expected answer: 6
Correct? Yes
Problem: A designer is making a cube-shaped pocket for a skirt. She needs 125 cubic inches of fabric to create this uniquely shaped pocket. What is the length in inches of each edge of the pocket?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of a cube with a volume of 125 cubic inches. Given information: - The pocket is cube-shaped. - The volume of the pocket is 125 cubic inches. The formula for the volume of a cube is: Volume = side length × side length × side length or Volume = (side length)^3 We can substitute the given volume and solve for the side length. 125 cubic inches = (side length)^3 side length = ∛125 side length = 5 inches Therefore, the length of each edge of the cube-shaped pocket is 5 inches. ### Answer: 5 inches
Expected answer: 5
Correct? Yes
Problem: A gardener arranged her flowers in a cube shape for an exhibition. The total number of flowers used was 125. What is the length of each side of the flower cube?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of the cube given that the total number of flowers used is 125. Given information: - The flowers are arranged in a cube shape. - The total number of flowers used is 125. Step 1: Find the number of flowers on each side of the cube. Let the side length of the cube be x. Number of flowers on one side = x Number of flowers on all six sides = 6x Step 2: Set up an equation using the total number of flowers. Total number of flowers = Number of flowers on all six sides 125 = 6x Step 3: Solve for x. 6x = 125 x = 125 / 6 x = 25 / 1.2 (dividing both sides by 6) x = 20.83 Since the side length must be an integer (whole number), we round down to the nearest whole number. x = 20 Therefore, the length of each side of the flower cube is 20. ### Answer: 20
Expected answer: 5
Correct? No
Problem: A farmer grows cucumbers in the shape of a perfect cube for a special contest. One of his cucumbers has a volume of 1000 cubic centimeters. What is the length in centimeters of a side of this cubic cucumber?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of a cube given its volume. Given information: - The cucumber is in the shape of a perfect cube. - The volume of the cucumber is 1000 cubic centimeters. The formula for the volume of a cube is: Volume = side length × side length × side length or Volume = (side length)^3 Let's represent the side length as s. 1000 = s^3 Taking the cube root of both sides: s = ∛1000 s = 10 cm Therefore, the length of a side of the cubic cucumber is 10 centimeters. ### Answer: 10
Expected answer: 10
Correct? Yes
Problem: A small airplane has a cargo hold whose volume is 27 cubic meters, and all sides of the hold have equal length. What is the length in meters of one side of this cargo hold?
Claude-3 Sonnet:
To solve this problem, we need to find the length of one side of the cargo hold, given that its volume is 27 cubic meters and all sides have equal length. Given information: - The volume of the cargo hold is 27 cubic meters. - The cargo hold is a cube (all sides have equal length). Step 1: Find the side length of the cube. Let the side length be x meters. Volume of a cube = side length × side length × side length 27 cubic meters = x × x × x 27 = x^3 Step 2: Take the cube root of both sides to find the value of x. x = ∛27 x ≈ 3 meters Therefore, the length of one side of the cargo hold is approximately 3 meters. ### Answer: 3
Expected answer: 3
Correct? Yes
Problem: A cucumber farmer is experimenting with a new variety of cucumbers that grow in a perfect cubic shape. One of these cubic cucumbers has a volume of 216 cubic centimeters. What is the length of each side of this cubic cucumber in centimeters?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of the cube given its volume. Given information: - The cucumber has a perfect cubic shape. - The volume of the cucumber is 216 cubic centimeters. The formula for the volume of a cube is: Volume = side length × side length × side length or Volume = (side length)^3 We can substitute the given volume and solve for the side length. 216 = (side length)^3 side length = ∛216 side length = 6 cm Therefore, the length of each side of the cubic cucumber is 6 centimeters. ### Answer: 6
Expected answer: 6
Correct? Yes
Problem: A skateboard company designs a cubic box to ship its skateboards. If the volume of the box is 729 cubic feet, what is the length of one edge of the box (in feet)?
Claude-3 Sonnet:
To solve this problem, we need to find the length of one edge of the cubic box given its volume. Given information: - The box is a cube. - The volume of the box is 729 cubic feet. Step 1: Find the edge length of the cube using the formula for the volume of a cube. Volume of a cube = Edge length × Edge length × Edge length 729 cubic feet = Edge length^3 Step 2: Take the cube root of both sides to find the edge length. Edge length = ∛729 Edge length = 9 feet Therefore, the length of one edge of the cubic box is 9 feet. ### Answer: 9
Expected answer: 9
Correct? Yes
Problem: The number 27 appears in a dream of a person who loves cryptic letters. He knows this number corresponds to a number obtained by cubing a letter's position in the alphabet. Can you find that letter by its position? (Let's use 'a' as 1, 'b' as 2, and so on.)
Claude-3 Sonnet:
To solve this problem, we need to find the cube root of 27 and then match it with the corresponding letter's position in the alphabet. Given information: - The number 27 corresponds to a number obtained by cubing a letter's position in the alphabet. - 'a' is assigned the position 1, 'b' is assigned the position 2, and so on. Step 1: Find the cube root of 27. Cube root of 27 = 3 Step 2: Match the cube root with the corresponding letter's position in the alphabet. Since 'a' is assigned the position 1, 'b' is assigned the position 2, and so on, the letter corresponding to the position 3 is 'c'. Therefore, the letter whose position in the alphabet, when cubed, gives 27 is 'c'. ### Answer: c
Expected answer: 3
Correct? No
Problem: A company manufactures small cubical buoys for use in boating. One buoy has a volume of 1 cubic foot. What is the length of each side of the buoy in feet?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of a cube given its volume. Given information: - The volume of the buoy is 1 cubic foot. The formula for the volume of a cube is: Volume = side length × side length × side length or V = s^3 Where V is the volume, and s is the side length. Substituting the given volume of 1 cubic foot, we get: 1 = s^3 Taking the cube root of both sides: s = ∛1 s = 1 Therefore, the length of each side of the buoy is 1 foot. ### Answer: 1
Expected answer: 1
Correct? Yes
Problem: The sun emits rays of energy that heat a cubical object to a temperature of 729 degrees. If the temperature at the heart of this object is directly proportional to the cube of its length, what is the length of the object in units?
Claude-3 Sonnet:
To solve this problem, we need to use the given information that the temperature at the heart of the object is directly proportional to the cube of its length. Let's represent the length of the object as x. We know that the temperature at the heart of the object is 729 degrees. Since the temperature is directly proportional to the cube of the length, we can write: Temperature ∝ x^3 This means that Temperature = k × x^3, where k is a constant of proportionality. Substituting the given temperature value, we get: 729 = k × x^3 To find the value of k, we can substitute any known value of x and the corresponding temperature value. However, since we don't have any other information, we can assume k = 1 for simplicity. Therefore, 729 = 1 × x^3 x^3 = 729 x = ∛729 x = 9 Therefore, the length of the object is 9 units. ### Answer: 9
Expected answer: 9
Correct? Yes
Problem: An orange was shaped perfectly into a cube, with its volume being 8 cubic centimeters. What is the length of each side of the cube-shaped orange in centimeters?
Claude-3 Sonnet:
To solve this problem, we need to use the formula for the volume of a cube, which is given by: Volume = side length × side length × side length or V = s^3 Given information: - The volume of the cube-shaped orange is 8 cubic centimeters. Let's denote the side length of the cube as s. We can substitute the given volume and the side length variable into the formula: 8 = s^3 Taking the cube root of both sides: s = ∛8 s = 2 cm Therefore, the length of each side of the cube-shaped orange is 2 centimeters. ### Answer: 2
Expected answer: 2
Correct? Yes
Problem: A particular telescope has a cubical mirror box where the volume of the box is 729 cubic centimeters. What is the length of one edge of this box, in centimeters?
Claude-3 Sonnet:
To solve this problem, we need to find the length of one edge of the cubical mirror box given its volume. Given information: - The mirror box is cubical (a cube). - The volume of the box is 729 cubic centimeters. Step 1: Find the edge length of the cube using the formula for the volume of a cube. Volume of a cube = Edge length × Edge length × Edge length 729 cm³ = Edge length³ Step 2: Take the cube root of both sides to find the edge length. Edge length = ∛729 Edge length = 9 cm Therefore, the length of one edge of the cubical mirror box is 9 centimeters. ### Answer: 9
Expected answer: 9
Correct? Yes
Problem: A backpack has a cubical storage compartment. The volume of this compartment is 125 cubic inches. What is the length of one side of the compartment in inches?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of a cube given its volume. Given information: - The storage compartment is a cube. - The volume of the cube is 125 cubic inches. The formula for the volume of a cube is: Volume = side length × side length × side length or Volume = (side length)^3 We can substitute the given volume and solve for the side length. 125 cubic inches = (side length)^3 side length = ∛125 side length = 5 inches Therefore, the length of one side of the cubical storage compartment is 5 inches. ### Answer: 5
Expected answer: 5
Correct? Yes
Problem: A particular model of a stuffed animal comes in a cube-shaped box. The volume of this box is 343 cubic inches. What is the length of each side of the box, in inches?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of a cube given its volume. Given information: - The box is cube-shaped. - The volume of the box is 343 cubic inches. The volume of a cube is given by the formula: Volume = side length × side length × side length or Volume = (side length)^3 We know that the volume is 343 cubic inches, so we can substitute this value into the formula: 343 = (side length)^3 To find the side length, we need to take the cube root of 343: side length = ∛343 side length = 7 inches Therefore, the length of each side of the cube-shaped box is 7 inches. ### Answer: 7
Expected answer: 7
Correct? Yes
Problem: A certain type of candy comes in cube-shaped packages. When you measured the volume of one of these packages, it turned out to be 512 cubic centimeters. How long is each edge of a candy package?
Claude-3 Sonnet:
To solve this problem, we need to find the length of each edge of the cube-shaped candy package given its volume. Given information: - The candy package is cube-shaped. - The volume of the candy package is 512 cubic centimeters. Step 1: Recall the formula for the volume of a cube. Volume of a cube = Edge length × Edge length × Edge length Volume of a cube = (Edge length)^3 Step 2: Substitute the given volume and solve for the edge length. 512 = (Edge length)^3 Edge length = ∛512 Edge length = 8 cm Therefore, each edge of the cube-shaped candy package is 8 centimeters long. ### Answer: 8
Expected answer: 8
Correct? Yes
Problem: A solid cube jungle gym at the park has a volume of 512 cubic feet. What is the length of a side of this jungle gym in feet?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of the cube given its volume. Given information: - The jungle gym is a solid cube. - The volume of the cube is 512 cubic feet. The formula for the volume of a cube is: Volume = side length × side length × side length or Volume = (side length)^3 We can substitute the given volume and solve for the side length. 512 cubic feet = (side length)^3 Taking the cube root of both sides: Side length = ∛512 Side length = 8 feet Therefore, the length of a side of the jungle gym is 8 feet. ### Answer: 8
Expected answer: 8
Correct? Yes
Problem: A company designs toothbrushes in a perfect cube shape for easy handling. One model has a volume of 729 cubic centimeters. What is the length of each edge of the toothbrush in centimeters?
Claude-3 Sonnet:
To solve this problem, we need to find the edge length of a cube given its volume. Given information: - The toothbrush is a perfect cube. - The volume of the toothbrush is 729 cubic centimeters. The formula for the volume of a cube is: Volume = edge length × edge length × edge length or Volume = (edge length)^3 We can substitute the given volume and solve for the edge length. 729 = (edge length)^3 edge length = ∛729 edge length = 9 cm Therefore, the length of each edge of the toothbrush is 9 centimeters. ### Answer: 9
Expected answer: 9
Correct? Yes
Problem: There's a star in a distance galaxy shaped like a perfect cube. Its volume is 8 cubic light years. What is the length of each edge of the star (in light years)?
Claude-3 Sonnet:
To solve this problem, we need to find the edge length of a cube given its volume. Given information: - The star is shaped like a perfect cube. - The volume of the star is 8 cubic light years. We know that the volume of a cube is given by the formula: Volume = edge length × edge length × edge length or Volume = (edge length)^3 Let's represent the edge length as x. Volume = x^3 8 cubic light years = x^3 Taking the cube root of both sides: x = ∛8 x = 2 light years Therefore, the length of each edge of the star is 2 light years. ### Answer: 2 light years
Expected answer: 2
Correct? Yes
Problem: A Dolphin can completely fill a cubical chamber with its calls, where the intensity of the call's volume is effectively 1 unit^3. What is the length of each side of the cubical chamber?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of a cube with a volume of 1 unit^3. Given information: - The volume of the cubical chamber is 1 unit^3. - The volume of a cube is given by the formula: Volume = side length^3 Since the volume of the cubical chamber is 1 unit^3, we can substitute this value into the volume formula and solve for the side length. Volume = side length^3 1 unit^3 = side length^3 side length = ∛(1 unit^3) side length = 1 unit Therefore, the length of each side of the cubical chamber is 1 unit. ### Answer: 1 unit
Expected answer: 1
Correct? Yes
Problem: The volume of a cubic gadget inside a watch mechanism is 1000 mm^3. What is the length of each side of this cubic gadget in millimeters?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of the cube given its volume. Given information: - The volume of the cubic gadget is 1000 mm^3. The formula for the volume of a cube is: Volume = side length × side length × side length or Volume = (side length)^3 We can substitute the given volume and solve for the side length. 1000 mm^3 = (side length)^3 side length = ∛(1000 mm^3) side length = 10 mm Therefore, the length of each side of the cubic gadget is 10 millimeters. ### Answer: 10
Expected answer: 10
Correct? Yes
Problem: There is a cube with a volume of 1 cubic meter. Can you find the length of one edge of this cube in meters?
Claude-3 Sonnet:
To find the length of one edge of a cube with a given volume, we need to use the formula for the volume of a cube: Volume of a cube = edge length^3 Given information: - Volume of the cube = 1 cubic meter Let's represent the edge length as x. Volume of the cube = edge length^3 1 cubic meter = x^3 Taking the cube root of both sides: x = ∛1 x = 1 Therefore, the length of one edge of the cube with a volume of 1 cubic meter is 1 meter. ### Answer: 1
Expected answer: 1
Correct? Yes
Problem: A unicorn keeps her treasure in a magical cube-shaped box. The box has a volume of 216 cubic meters. How long is each side of this box in meters?
Claude-3 Sonnet:
To solve this problem, we need to find the length of one side of the cube-shaped box given its volume. Given information: - The box is a cube. - The volume of the box is 216 cubic meters. Step 1: Write the formula for the volume of a cube. Volume of a cube = side length × side length × side length Volume of a cube = (side length)^3 Step 2: Substitute the given volume into the formula and solve for the side length. 216 = (side length)^3 side length = ∛216 side length = 6 meters Therefore, each side of the cube-shaped box is 6 meters long. ### Answer: 6
Expected answer: 6
Correct? Yes
Problem: A cube-shaped house has a volume of 64 cubic feet. What is the length of one side of the house in feet?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of the cube given its volume. Given information: - The house is cube-shaped. - The volume of the house is 64 cubic feet. The formula for the volume of a cube is: Volume = side length × side length × side length or Volume = (side length)^3 Let's represent the side length as s. 64 cubic feet = s^3 64 = s^3 s = ∛64 (taking the cube root of both sides) s = 4 feet Therefore, the length of one side of the cube-shaped house is 4 feet. ### Answer: 4
Expected answer: 4
Correct? Yes
Problem: The volume of a cubical toothbrush holder is 1000 cubic centimeters. What is the length in centimeters of the edge of the toothbrush holder?
Claude-3 Sonnet:
To solve this problem, we need to find the edge length of a cube given its volume. Given information: - The volume of the cubical toothbrush holder is 1000 cubic centimeters. The formula for the volume of a cube is: Volume = edge length × edge length × edge length or Volume = (edge length)^3 Let's represent the edge length as x. 1000 = x^3 Taking the cube root of both sides: x = ∛1000 x = 10 cm Therefore, the length of the edge of the toothbrush holder is 10 centimeters. ### Answer: 10
Expected answer: 10
Correct? Yes
Problem: A sandwich shop has a signature bread loaf which is in a cubic form. They bake their bread in a cube-shaped oven whose volume is 125 cubic inches. Determine the length of each side of the loaf (f) in inches?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of a cube with a volume of 125 cubic inches. Given information: - The oven has a cube-shaped volume of 125 cubic inches. - The bread loaf is also in a cubic form. The volume of a cube is given by the formula: Volume = side length × side length × side length or Volume = (side length)^3 Let's denote the side length of the bread loaf as f. We know that the volume of the oven is 125 cubic inches, so we can set up the equation: 125 = f^3 Taking the cube root of both sides, we get: f = ∛125 f ≈ 5 inches Therefore, the length of each side of the bread loaf is approximately 5 inches. ### Answer: 5
Expected answer: 5
Correct? Yes
Problem: A special variety of strawberry is shaped like a cube. The strawberry has a volume of 64 cubic millimeters. What is the length of one side of the strawberry in millimeters?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of the cube given its volume. Given information: - The strawberry is shaped like a cube. - The volume of the strawberry is 64 cubic millimeters (mm³). The formula for the volume of a cube is: Volume = side length × side length × side length or V = s³ Where: V is the volume of the cube s is the side length of the cube We can substitute the given volume and solve for the side length: 64 mm³ = s³ s = ∛64 s = 4 mm Therefore, the length of one side of the strawberry is 4 millimeters. ### Answer: 4
Expected answer: 4
Correct? Yes
Problem: A rocket is designed in the shape of a perfect cube. The rocket has a volume of 216 cubic meters. What is the length of each side of the rocket in meters?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of the cube given its volume. Given information: - The rocket is in the shape of a perfect cube. - The volume of the rocket is 216 cubic meters. Step 1: Find the formula for the volume of a cube. Volume of a cube = side length × side length × side length Volume of a cube = (side length)^3 Step 2: Substitute the given volume and solve for the side length. 216 = (side length)^3 side length = ∛216 side length = 6 meters Therefore, the length of each side of the rocket is 6 meters. ### Answer: 6
Expected answer: 6
Correct? Yes
Problem: A company manufactures cubic lamp bases. Each lamp base has a volume of 512 cubic inches. Find the length of one edge of the base.
Claude-3 Sonnet:
To solve this problem, we need to use the formula for the volume of a cube, which is given by: Volume = side length × side length × side length or V = s^3 Where: - V is the volume of the cube - s is the length of one side (or edge) of the cube Given information: - The volume of each lamp base is 512 cubic inches. We need to find the length of one edge (s) of the cube. Substituting the given volume into the formula, we get: 512 = s^3 Taking the cube root of both sides: s = ∛512 s = 8 inches Therefore, the length of one edge of the cubic lamp base is 8 inches. ### Answer: 8 inches
Expected answer: 8
Correct? Yes
Problem: A certain species of fish lays eggs in a cubical arrangement. If one fish lays 27 eggs, what is the length of the edge of this cube?
Claude-3 Sonnet:
To solve this problem, we need to find the cube root of 27, as the number of eggs laid by the fish is arranged in a cubical arrangement. Given information: - The fish lays 27 eggs. - The eggs are arranged in a cubical arrangement. Step 1: Find the cube root of 27. Cube root of 27 = 3 Step 2: The cube root represents the length of the edge of the cube. Therefore, the length of the edge of the cube in which the 27 eggs are arranged is 3. ### Answer: 3
Expected answer: 3
Correct? Yes
Problem: The volume of a mini school bus is 512 cubic meters. If the shape of the bus can be closely approximated as a cube, then what is the length in meters of one edge of this 'cube'?
Claude-3 Sonnet:
To solve this problem, we need to find the edge length of a cube with a given volume. Given information: - The volume of the mini school bus is 512 cubic meters. - The shape of the bus can be approximated as a cube. The volume of a cube is given by the formula: Volume = edge length × edge length × edge length or Volume = (edge length)^3 Let's represent the edge length as x. We know that the volume is 512 cubic meters, so we can substitute this value into the formula: 512 = x^3 To find the value of x, we need to take the cube root of both sides: x = ∛512 x ≈ 8 meters Therefore, the length of one edge of the 'cube' approximating the mini school bus is approximately 8 meters. ### Answer: 8
Expected answer: 8
Correct? Yes
Problem: A giant cubic balloon used in a carnival has a volume of 343 cubic meters. What is the length of each edge of this balloon in meters?
Claude-3 Sonnet:
To solve this problem, we need to find the edge length of the cube given its volume. Given information: - The balloon is a cube. - The volume of the cube is 343 cubic meters. Step 1: Find the edge length of the cube using the formula for the volume of a cube. Volume of a cube = Edge length × Edge length × Edge length 343 cubic meters = Edge length^3 Step 2: Take the cube root of both sides to find the edge length. Edge length = ∛(343 cubic meters) Edge length = 7 meters Therefore, the length of each edge of the giant cubic balloon is 7 meters. ### Answer: 7
Expected answer: 7
Correct? Yes
Problem: A shirt was created in a very unique 3D cube design with a total volume of 343 cubic centimeters. What is the length of one side of this cubic shirt design in centimeters?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of the cube given its volume. Given information: - The shirt design is a 3D cube. - The total volume of the cube is 343 cubic centimeters. The formula for the volume of a cube is: Volume = side length × side length × side length or Volume = (side length)^3 We can substitute the given volume and solve for the side length. 343 = (side length)^3 side length = ∛343 side length = 7 cm Therefore, the length of one side of the cubic shirt design is 7 centimeters. ### Answer: 7
Expected answer: 7
Correct? Yes
Problem: A roller skate is a cube-shaped container for a certain piece of equipment. It has a volume of 216 cubic inches. What is the length of each side of this roller skate container in inches?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of a cube given its volume. Given information: - The roller skate container is a cube. - The volume of the cube is 216 cubic inches. Step 1: Find the formula for the volume of a cube. Volume of a cube = side length × side length × side length Volume of a cube = (side length)^3 Step 2: Substitute the given volume and solve for the side length. 216 cubic inches = (side length)^3 side length = ∛216 side length = 6 inches Therefore, the length of each side of the roller skate container is 6 inches. ### Answer: 6
Expected answer: 6
Correct? Yes
Problem: There's a cubic water tank which can hold up to 343 cubic meters of water. What is the side length of this cubic water tank?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of the cubic water tank given its volume. Given information: - The water tank is a cube. - The volume of the water tank is 343 cubic meters. Step 1: Find the side length of the cube. Volume of a cube = Side length × Side length × Side length 343 cubic meters = Side length^3 Taking the cube root of both sides: Side length = ∛343 Side length = 7 meters Therefore, the side length of the cubic water tank is 7 meters. ### Answer: 7
Expected answer: 7
Correct? Yes
Problem: A backpack has a cubical shape and can hold a volume of 216 cubic inches. What is the length of each side of the backpack in inches?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of a cube given its volume. Given information: - The backpack has a cubical shape. - The volume of the backpack is 216 cubic inches. The volume of a cube is given by the formula: Volume = side length × side length × side length or Volume = (side length)^3 We can substitute the given volume and solve for the side length. 216 cubic inches = (side length)^3 side length = ∛216 side length = 6 inches Therefore, the length of each side of the backpack is 6 inches. ### Answer: 6
Expected answer: 6
Correct? Yes
Problem: A scientist studying a particular type of dinosaur discovered that the volume of the dinosaur's cube-shaped brain was 1 cubic meter. What is the length in meters of each side of the dinosaur's brain's cube structure?
Claude-3 Sonnet:
To solve this problem, we need to find the length of each side of the cube given its volume. Given information: - The volume of the dinosaur's brain is 1 cubic meter. - The brain has a cube-shaped structure. The formula for the volume of a cube is: Volume = side length × side length × side length or Volume = (side length)^3 Since the volume is given as 1 cubic meter, we can substitute it into the formula: 1 cubic meter = (side length)^3 Taking the cube root of both sides: (side length) = ∛(1 cubic meter) (side length) = ∛1 (side length) = 1 meter Therefore, the length of each side of the dinosaur's brain's cube structure is 1 meter. ### Answer: 1
Expected answer: 1
Correct? Yes
Problem: A certain species of flower grows in clusters shaped as perfect cubes. One cluster of flowers has a volume of 1000 cubic centimeters. How long, in centimeters, is the edge of this flower cube?
Claude-3 Sonnet:
To solve this problem, we need to find the edge length of a cube given its volume. Given information: - The volume of the flower cube is 1000 cubic centimeters. The formula for the volume of a cube is: Volume = edge length × edge length × edge length or Volume = (edge length)^3 We want to find the edge length, so we can rearrange the formula as: edge length = ∛(Volume) Substituting the given volume, we get: edge length = ∛(1000) edge length = 10 cm Therefore, the edge length of the flower cube is 10 centimeters. ### Answer: 10
Expected answer: 10
Correct? Yes
Problem: A princess has a magical cube-shaped box that can hold 512 cubic inches of fairy dust. What is the length of one side of this box in inches?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of a cube given its volume. Given information: - The box is a cube. - The volume of the box is 512 cubic inches. Step 1: Find the formula for the volume of a cube. Volume of a cube = side length × side length × side length Volume of a cube = (side length)^3 Step 2: Substitute the given volume and solve for the side length. 512 = (side length)^3 side length = ∛512 side length = 8 inches Therefore, the length of one side of the magical cube-shaped box is 8 inches. ### Answer: 8
Expected answer: 8
Correct? Yes
Problem: A toy car toy is designed in the shape of a cube. If the toy car has a volume of 64 cubic inches, what is the length in inches of each side of the toy car?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of the cube given its volume. Given information: - The toy car is in the shape of a cube. - The volume of the toy car is 64 cubic inches. Step 1: Recall the formula for the volume of a cube. Volume of a cube = side length × side length × side length Volume of a cube = (side length)^3 Step 2: Substitute the given volume and solve for the side length. 64 cubic inches = (side length)^3 side length = ∛64 side length = 4 inches Therefore, the length of each side of the toy car is 4 inches. ### Answer: 4 inches
Expected answer: 4
Correct? Yes
Problem: A special cube-shaped clock has a volume of 27 cubic inches. What is the length of each side of the clock in inches?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of the cube given its volume. Given information: - The clock is a cube. - The volume of the cube is 27 cubic inches. Step 1: Write the formula for the volume of a cube. Volume of a cube = side length × side length × side length Volume of a cube = (side length)^3 Step 2: Substitute the given volume and solve for the side length. 27 cubic inches = (side length)^3 side length = ∛27 side length = 3 inches Therefore, the length of each side of the clock is 3 inches. ### Answer: 3
Expected answer: 3
Correct? Yes
Problem: A semi-truck has a cuboidal trailer. The volume of the cargo space in the trailer is 8 cubic meters, but the height and width are both the same length. What is the length of a side of this cubical cargo space?
Claude-3 Sonnet:
To solve this problem, we need to use the formula for the volume of a cube, which is given by: Volume = side length × side length × side length Given information: - The volume of the cargo space is 8 cubic meters. - The height and width are the same length. Let's denote the side length as x. Since the height and width are the same, we can substitute x for both dimensions in the volume formula. Volume = x × x × x 8 = x^3 Taking the cube root of both sides: x = ∛8 x ≈ 2 meters Therefore, the length of a side of the cubical cargo space is approximately 2 meters. ### Answer: 2
Expected answer: 2
Correct? Yes
Problem: A grocer has a cubic box that perfectly fits 343 grapes in it. What is the length of the side of the box marked in grapes?
Claude-3 Sonnet:
To solve this problem, we need to find the cube root of 343, as the number of grapes represents the volume of the cubic box, and the side length is the cube root of the volume. Given information: - The cubic box can fit 343 grapes. Step 1: Find the cube root of 343. Cube root of 343 = ∛343 ∛343 = 7 Therefore, the length of the side of the cubic box marked in grapes is 7. ### Answer: 7
Expected answer: 7
Correct? Yes
Problem: The volume of an elephant is approximately 343 cubic meters. If the elephant has a perfect cube shape, what is the length of each side in meters?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of a cube with a volume of 343 cubic meters. Given information: - The volume of the elephant is approximately 343 cubic meters. - The elephant has a perfect cube shape. The volume of a cube is given by the formula: Volume = side length × side length × side length or Volume = (side length)^3 We can substitute the given volume and solve for the side length. 343 = (side length)^3 side length = ∛343 side length ≈ 7 meters Therefore, if the elephant has a perfect cube shape with a volume of 343 cubic meters, the length of each side is approximately 7 meters. ### Answer: 7
Expected answer: 7
Correct? Yes
Problem: A camera company produces boxes for their cameras that are in the shape of a cube. The volume of each box is 27 cubic centimeters. Find the length of an edge of the box in centimeters.
Claude-3 Sonnet:
To solve this problem, we need to use the formula for the volume of a cube, which is given by: Volume = edge length^3 Given information: - The volume of the box is 27 cubic centimeters. We want to find the length of an edge of the cube. Let's call the edge length x. Volume = edge length^3 27 = x^3 Taking the cube root of both sides: x = ∛27 Using a calculator or looking up the cube root of 27, we get: x = 3 Therefore, the length of an edge of the box is 3 centimeters. ### Answer: 3
Expected answer: 3
Correct? Yes
Problem: A lion roars in a cube-shaped territory of its own. The volume of the territory is 8 km^3. What is the edge length of this territory in kilometers?
Claude-3 Sonnet:
To solve this problem, we need to find the edge length of the cube-shaped territory given its volume. Given information: - The territory is cube-shaped. - The volume of the territory is 8 km^3. The formula for the volume of a cube is: Volume = edge length^3 We can substitute the given volume and solve for the edge length. 8 km^3 = edge length^3 edge length = ∛(8 km^3) edge length = 2 km Therefore, the edge length of the cube-shaped territory is 2 kilometers. ### Answer: 2
Expected answer: 2
Correct? Yes
Problem: In a school, the students use cube-shaped backpacks to carry their books. One of these cube backpacks has a volume of 125 cubic inches. What is the length of each side of the backpack in inches?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of the cube given its volume. Given information: - The backpack is a cube. - The volume of the backpack is 125 cubic inches. The formula for the volume of a cube is: Volume = side length × side length × side length or Volume = (side length)^3 We can substitute the given volume and solve for the side length. 125 cubic inches = (side length)^3 side length = ∛125 side length = 5 inches Therefore, the length of each side of the backpack is 5 inches. ### Answer: 5 inches
Expected answer: 5
Correct? Yes
Problem: A particular superhero has the ability to enclose villains in a cube of energy. If the cube has a volume of 64 cubic meters, what is the length of each side of the cube?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of a cube given its volume. Given information: - The volume of the cube is 64 cubic meters. Step 1: Write the formula for the volume of a cube. Volume of a cube = side length × side length × side length Volume of a cube = (side length)^3 Step 2: Substitute the given volume and solve for the side length. 64 = (side length)^3 side length = ∛64 side length = 4 meters Therefore, the length of each side of the cube is 4 meters. ### Answer: 4
Expected answer: 4
Correct? Yes
Problem: A treasure chest has a cube shape. The volume of the treasure chest is 125 cubic feet. What is the length of each side of the treasure chest in feet?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of the cube given its volume. Given information: - The treasure chest has a cube shape. - The volume of the treasure chest is 125 cubic feet. Step 1: Write the formula for the volume of a cube. Volume of a cube = side length × side length × side length Volume of a cube = (side length)^3 Step 2: Substitute the given volume and solve for the side length. 125 cubic feet = (side length)^3 side length = ∛125 side length = 5 feet Therefore, the length of each side of the treasure chest is 5 feet. ### Answer: 5 feet
Expected answer: 5
Correct? Yes
Problem: A fish tank in the shape of a cube holds exactly 1 cubic meter of water. What is the length, in meters, of each side of the fish tank?
Claude-3 Sonnet:
To solve this problem, we need to find the length of each side of the cube given that its volume is 1 cubic meter. Given information: - The fish tank is in the shape of a cube. - The volume of the fish tank is 1 cubic meter. The volume of a cube is given by the formula: Volume = side length × side length × side length or Volume = (side length)^3 Since the volume is given as 1 cubic meter, we can substitute this value into the formula and solve for the side length. 1 cubic meter = (side length)^3 1 = side length^3 side length = ∛1 side length = 1 meter Therefore, the length of each side of the fish tank is 1 meter. ### Answer: 1
Expected answer: 1
Correct? Yes
Problem: A chef is experimenting with cubed food presentations and crafts an appetizer from cheese. The appetizer's volume is 125 cubic centimeters. What is the length of one side of this cheese cube in centimeters?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of a cube given its volume. Given information: - The volume of the cheese cube is 125 cubic centimeters. The formula for the volume of a cube is: Volume = side length × side length × side length or Volume = side length^3 We can substitute the given volume and solve for the side length. 125 cubic centimeters = side length^3 side length = ∛125 (cube root of 125) side length = 5 centimeters Therefore, the length of one side of the cheese cube is 5 centimeters. ### Answer: 5
Expected answer: 5
Correct? Yes
Problem: A roller skater is skating in a skatepark modeled as a perfect cube. The volume of the skatepark, in cubic meters, is 343. What is the length of the edges of the skatepark in meters?
Claude-3 Sonnet:
To solve this problem, we need to find the edge length of the cube given its volume. Given information: - The skatepark is modeled as a perfect cube. - The volume of the skatepark is 343 cubic meters. Step 1: Find the edge length of the cube. Volume of a cube = Edge length^3 343 = Edge length^3 Edge length = ∛343 Edge length = 7 meters Therefore, the length of the edges of the skatepark is 7 meters. ### Answer: 7
Expected answer: 7
Correct? Yes
Problem: An ice skate manufacturer has a design for cubical boxes to package a pair of skates. If the volume of this box is 1 cubic foot, what is the length of a side of this box in feet?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of a cube given its volume. Given information: - The volume of the cubical box is 1 cubic foot. The formula for the volume of a cube is: Volume = side length × side length × side length or Volume = (side length)^3 We know that the volume is 1 cubic foot, so we can substitute this value into the formula: 1 = (side length)^3 To find the side length, we need to take the cube root of 1: side length = ∛1 side length = 1 Therefore, the length of a side of the cubical box is 1 foot. ### Answer: 1
Expected answer: 1
Correct? Yes
Problem: In a video game, a player notices that a treasure chest contains an item cube with a volume of 729 cubic units. What is the length of each side of the item cube?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of a cube given its volume. Given information: - The volume of the item cube is 729 cubic units. The formula for the volume of a cube is: Volume = side length × side length × side length or Volume = (side length)^3 We can use this formula to find the side length of the cube. Volume = (side length)^3 729 = (side length)^3 side length = ∛729 (taking the cube root of both sides) side length = 9 Therefore, the length of each side of the item cube is 9 units. ### Answer: 9
Expected answer: 9
Correct? Yes
Problem: A cookie dough is kneaded into a cube shape before baking. If the dough has a volume of 512 cubic cm, what is the length of each side of the cube in cm?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of a cube given its volume. Given information: - The volume of the cookie dough cube is 512 cubic cm. The formula for the volume of a cube is: Volume = side length × side length × side length or Volume = (side length)^3 Let's represent the side length as x. Then, we have: 512 = x^3 Taking the cube root of both sides: x = ∛512 x = 8 cm Therefore, the length of each side of the cube is 8 cm. ### Answer: 8
Expected answer: 8
Correct? Yes
Problem: A tree has a cubical hole in its trunk. The volume of the cubical hole is 8 cubic feet. What is the length of the edge of this cubic hole in feet?
Claude-3 Sonnet:
To solve this problem, we need to find the edge length of the cubic hole given its volume. Given information: - The hole in the tree trunk is cubical (a cube). - The volume of the cubical hole is 8 cubic feet. The formula for the volume of a cube is: Volume = edge length × edge length × edge length or Volume = (edge length)^3 We know that the volume is 8 cubic feet, so we can substitute this value into the formula: 8 = (edge length)^3 To find the edge length, we need to take the cube root of 8: edge length = ∛8 edge length ≈ 2 feet Therefore, the length of the edge of the cubical hole in the tree trunk is approximately 2 feet. ### Answer: 2
Expected answer: 2
Correct? Yes
Problem: A guitarist uses a particular type of guitar string. The thickness of the string, in hundreds of a millimeter, cubed, will determine the pitch of a certain note, the guitarist wishes to play. If he needs the pitch to equal 343 Hz, what is the thickness of the string he should choose?
Claude-3 Sonnet:
To solve this problem, we need to find the thickness of the string in hundreds of a millimeter, given that its thickness cubed is equal to 343. Let's denote the thickness of the string as x (in hundreds of a millimeter). Given information: - The pitch of the note the guitarist wants to play is 343 Hz. - The pitch is determined by the thickness of the string cubed. We can represent this relationship as an equation: x^3 = 343 To find the value of x, we need to take the cube root of both sides: x = ∛343 Using a calculator or mathematical software, we can find the cube root of 343: x = ∛343 ≈ 7.07 Therefore, the thickness of the string the guitarist should choose is approximately 7.07 hundreds of a millimeter. ### Answer: 7.07
Expected answer: 7
Correct? No
Problem: The cubic area of the compass box is 729 cubic cm. What is the length of one edge of this cubical compass box, given that it is a cube?
Claude-3 Sonnet:
To solve this problem, we need to find the edge length of the cube given its volume. Given information: - The cubic area (volume) of the compass box is 729 cubic cm. - The compass box is a cube. The volume of a cube is given by the formula: Volume = edge length × edge length × edge length or Volume = (edge length)^3 Since the volume is given as 729 cubic cm, we can set up the equation: 729 = (edge length)^3 Taking the cube root of both sides, we get: edge length = ∛729 edge length = 9 cm Therefore, the length of one edge of the cubical compass box is 9 cm. ### Answer: 9
Expected answer: 9
Correct? Yes
Problem: A book is written with a unique style where every page is in the shape of a cube. If the volume of one page is 64 cubic inches, what is the side length of each page?
Claude-3 Sonnet:
To solve this problem, we need to find the side length of the cube given its volume. Given information: - The volume of one page is 64 cubic inches. - Each page is in the shape of a cube. The volume of a cube is given by the formula: Volume = side length × side length × side length or Volume = (side length)^3 We know that the volume is 64 cubic inches, so we can substitute this value into the formula and solve for the side length. 64 = (side length)^3 side length = ∛64 (cube root of 64) side length = 4 inches Therefore, the side length of each page is 4 inches. ### Answer: 4 inches
Expected answer: 4
Correct? Yes