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 Opus:

Let's approach this step by step:

1) We know that the sandbox is in the shape of a cube. In a cube, all sides have equal length. Let's call the length of each side $x$ feet.

2) The volume of a cube is given by the cube of the length of its side. In this case, the volume is 512 cubic feet. We can express this as an equation:

   $x^3 = 512$

3) To find the length of each side, we need to find the cube root of 512. 

   $x = \sqrt[3]{512}$

4) The cube root of 512 is 8.

   $x = 8$

Therefore, the length of each side of the sandbox is 8 feet.

### Answer: 8 feet.

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 Opus:

Let's approach this step by step:

1) We know that the fireplace is a cube. In a cube, all sides have equal length. Let's call the length of each side $x$ feet.

2) The volume of a cube is given by the cube of its side length. So, if the volume is 729 cubic feet and the side length is $x$ feet, we can write:

   $x^3 = 729$

3) To find $x$, we need to find the cube root of 729. 

   $x = \sqrt[3]{729}$

4) The cube root of 729 is 9.

   $x = 9$

Therefore, the length of each side of the fireplace is 9 feet.

### Answer: 9 feet.

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 Opus:

Let's solve this step by step:

1) We know that the barn is a cube. In a cube, all sides have equal length. Let's call the length of each side $x$ meters.

2) The volume of a cube is given by the cube of the length of its side. So, if the volume is 1000 cubic meters and the length of each side is $x$ meters, we can write:

   $x^3 = 1000$

3) To find $x$, we need to find the cube root of 1000. 

   $x = \sqrt[3]{1000}$

4) The cube root of 1000 is 10.

   $x = 10$

Therefore, the length of each side of the barn is 10 meters.

### Answer: 10 meters.

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 Opus:

Let's approach this step by step:

1) We know that the tank is a cube. In a cube, all sides are equal in length. Let's call the length of one side $x$.

2) The volume of a cube is given by the cube of the length of its side. In other words, if the length of a side is $x$, the volume is $x^3$.

3) We are told that the volume of the tank is 512 m^3. This means:

   $x^3 = 512$

4) To find $x$, we need to find the cube root of 512. The cube root of a number is a value that, when cubed, gives the number. 

5) The cube root of 512 is 8, because $8^3 = 8 \times 8 \times 8 = 512$.

   So, $x = 8$.

6) Therefore, the length of each side of the tank is 8 meters.

### Answer: 8 meters.

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 Opus:

Let's approach this step by step:

1) We know that the volume of a cube is given by the formula:

   Volume = side length^3

2) In this case, the volume is 64 km^3. So we can write:

   64 = side length^3

3) To find the side length, we need to find the cube root of 64. The cube root of a number is a value that, when cubed (raised to the power of 3), gives the number.

4) The cube root of 64 is 4, because 4^3 = 4 × 4 × 4 = 64.

Therefore, the length of each side of the cuboidal mountain is 4 km.

### Answer: 4 km

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 Opus:

Let's approach this step by step:

1) We know that the crater is in the shape of a cube. In a cube, the length, width, and height are all equal. Let's call this length $l$.

2) We also know that the volume of the crater is 512 cubic kilometers. The volume of a cube is given by the formula:

   Volume = $l^3$

3) Substituting the given volume:

   512 = $l^3$

4) To find the length of each edge, we need to find the cube root of 512:

   $l = \sqrt[3]{512}$

5) The cube root of 512 is 8.

   $l = 8$

Therefore, the length of each edge of the crater is 8 kilometers.

### Answer: 8 kilometers.

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 Opus:

Let's approach this step by step:

1) We know that the baby is building a cube. A cube has equal length, width, and height.

2) Let's say the length of one side of the cube is x blocks.

3) In a cube, the volume is equal to the cube of the length of one side. In other words, volume = x³.

4) We are told that the total number of blocks is 343. This means the volume of the cube is 343 blocks.

5) So, we can set up an equation: x³ = 343

6) To solve for x, we need to find the cube root of 343.

7) The cube root of 343 is 7, because 7 * 7 * 7 = 343.

8) Therefore, x = 7.

### 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 Opus:

Let's approach this step by step:

1) We know that the container is a cube. In a cube, all sides have equal length. Let's call the length of one side $s$.

2) The volume of a cube is given by the cube of the length of its side. In other words, if the length of each side is $s$, then the volume is $s^3$.

3) We are given that the volume of the substance in the cubic container is 512 cubic feet. This means:

   $s^3 = 512$

4) To find $s$, we need to find the cube root of 512. The cube root of a number is a value that, when cubed (raised to the power of 3), gives the number. 

5) The cube root of 512 is 8, because $8^3 = 8 \times 8 \times 8 = 512$.

Therefore, the length of one side of the cubic container is 8 feet.

### Answer: 8 feet.

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 Opus:

Let's solve this step by step:

1) We know that the volume of a cube is given by the formula:

   Volume = edge length^3

2) In this case, the volume is 512 mm^3. Let's call the edge length x. So we can write:

   512 = x^3

3) To find x, we need to take the cube root of both sides:

   ∛512 = ∛(x^3)

   ∛512 = x

4) The cube root of 512 is 8, because 8 * 8 * 8 = 512.

   So, x = 8.

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 Opus:

Let's approach this step by step:

1) We know that the volume of a cube is given by the formula:

   Volume = side length^3

2) In this case, the volume is 8 cubic cm. So we can write:

   8 = side length^3

3) To find the side length, we need to take the cube root of both sides:

   ∛8 = ∛(side length^3)

   ∛8 = side length

4) The cube root of 8 is 2, because 2^3 = 8.

   2 = side length

Therefore, the length of each side of the cube is 2 cm.

### Answer: 2 cm.

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 Opus:

Let's approach this step by step:

1) We know that the volume of a cube is given by the formula:

   Volume = edge length^3

2) In this case, the volume is 27 cubic kilometers. Let's denote the edge length by 'x'. Then:

   27 = x^3

3) To find the value of x, we need to find the cube root of 27. 

   x = ∛27

4) The cube root of 27 is 3, because 3 * 3 * 3 = 27.

   x = 3

5) Therefore, the length of one edge of the cube is 3 kilometers.

### Answer: 3 kilometers.

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 Opus:

Let's approach this step by step:

1) We know that the box is a cube. In a cube, all sides have equal length. Let's call the length of each side $x$ inches.

2) The volume of a cube is given by the cube of the length of its side. In other words, if the length of each side is $x$, then the volume is $x^3$.

3) We are given that the volume of the box is 1 cubic inch. Therefore, we can write the equation:

   $x^3 = 1$

4) To solve for $x$, we need to find the cube root of both sides:

   $\sqrt[3]{x^3} = \sqrt[3]{1}$

   $x = 1$

Therefore, the length of each side of the box is 1 inch.

### Answer: 1 inch

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 Opus:

Let's approach this step by step:

1) We know that the flower is in the shape of a cube. Let's say the length of one side of the cube is $x$ centimeters.

2) The volume of a cube is given by the cube of its side length. So, the volume of the flower is $x^3$ cubic centimeters.

3) We are told that each cubic centimeter of the flower holds 1 milliliter of nectar. Therefore, the volume in cubic centimeters is numerically equal to the volume in milliliters.

4) We are also told that this particular flower holds 125 milliliters of nectar. Therefore, the volume of the flower is 125 cubic centimeters.

5) We can now set up an equation:

   $x^3 = 125$

6) To solve for $x$, we need to take the cube root of both sides:

   $\sqrt[3]{x^3} = \sqrt[3]{125}$
   
   $x = 5$

Therefore, the length of one side of this cubic flower is 5 centimeters.

### Answer: 5 centimeters.

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 Opus:

Let's solve this step by step:

1) We know that the volume of a cube is given by the formula:

   Volume = side length³

2) In this case, the volume is 125 cubic meters. So we can write:

   125 = side length³

3) To find the side length, we need to find the cube root of 125. 

4) The cube root of 125 is 5, because 5³ = 5 × 5 × 5 = 125.

Therefore, the length of one side of the tree stump is 5 meters.

### Answer: 5 meters.

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 Opus:

Let's solve this step by step:

1) We know that the volume of a cube is given by the formula:

   Volume = edge length^3

2) In this case, the volume is 1000 cubic millimeters. Let's call the edge length x millimeters. So we have:

   1000 = x^3

3) To find x, we need to take the cube root of both sides:

   ∛1000 = ∛(x^3)

   ∛1000 = x

4) The cube root of 1000 is 10.

   10 = x

So, the length of each edge of the bracelet is 10 millimeters.

### Answer: 10 millimeters.

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 Opus:

Let's approach this step by step:

1) We know that the soap is in the shape of a cube. In a cube, all sides are equal in length. Let's call the length of each side $x$ cm.

2) The volume of a cube is given by the cube of the length of its side. In other words, if the length of each side is $x$ cm, then the volume is $x^3$ cubic cm.

3) We are given that the volume of the soap is 1 cubic cm. Therefore, we can write an equation:

   $x^3 = 1$

4) To solve for $x$, we need to find the cube root of both sides:

   $\sqrt[3]{x^3} = \sqrt[3]{1}$
   
   $x = 1$

Therefore, the length of each side of the soap 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 Opus:

Let's solve this step by step:

1) We know that the volume of a cube is given by the formula:

   Volume = edge length³

2) We are given that the volume of the cubic yew tree is 216 cubic meters. Let's call the edge length x meters. Then:

   216 = x³

3) To find x, we need to find the cube root of 216. 

   x = ∛216

4) The cube root of 216 is 6, because 6 * 6 * 6 = 216.

   x = 6

Therefore, the length of each edge of the yew tree is 6 meters.

### Answer: 6 meters.

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 Opus:

Let's approach this step by step:

1) We know that the blanket is folded into a perfect cube. In a cube, all sides have equal length. Let's call the length of each side $x$ cm.

2) The volume of a cube is given by the cube of the length of its side. So, if the volume is 1000 cubic cm and the length of each side is $x$ cm, we can write:

   $x^3 = 1000$

3) To find $x$, we need to find the cube root of 1000:

   $x = \sqrt[3]{1000}$

4) The cube root of 1000 is 10.

   $x = 10$

5) Therefore, the length of each side of the blanket is 10 cm.

### Answer: 10 cm

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 Opus:

Let's solve this step by step:

1) We know that the box is a cube. In a cube, all sides have equal length. Let's call the length of each side $x$.

2) The volume of a cube is given by the cube of the length of its side. In other words, if the length of each side is $x$, then the volume is $x^3$.

3) We are given that the volume of the box is 125 cubic centimeters. Therefore:

   $x^3 = 125$

4) To find $x$, we need to find the cube root of 125. The cube root of a number is a value that, when cubed, gives the number.

5) The cube root of 125 is 5, because $5^3 = 5 \times 5 \times 5 = 125$.

Therefore, the length of each side of the box is 5 centimeters.

### Answer: 5 centimeters.

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 Opus:

Let's solve this step by step:

1) We know that the volume of a cube is given by the formula:

   Volume = side length × side length × side length
   
   or
   
   Volume = side length^3

2) We are given that the volume is 125 cubic centimeters. Let's call the side length s. Then:

   125 = s^3

3) To find s, we need to find the cube root of 125. The cube root of a number is a value that, when cubed (raised to the power of 3), gives the number.

4) The cube root of 125 is 5, because 5^3 = 5 × 5 × 5 = 125.

   So, s = 5.

Therefore, the length of each side of the perfume bottle is 5 centimeters.

### Answer: 5 centimeters.

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 Opus:

Let's solve this step by step:

1) We know that the building is a cube, which means all its sides are equal. Let's call the length of one side $x$.

2) The volume of a cube is given by the formula: $V = x^3$, where $V$ is the volume and $x$ is the length of one side.

3) We are given that the volume of the building is 216 cubic meters. So, we can write:

   $216 = x^3$

4) To find $x$, we need to find the cube root of 216. 

   $x = \sqrt[3]{216}$

5) The cube root of 216 is 6.

   $x = 6$

Therefore, the length of one side of the building is 6 meters.

### Answer: 6 meters.

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 Opus:

Let's approach this step by step:

1) We know that the chair is in the shape of a cube. In a cube, all sides are equal in length. Let's call the length of each side $x$.

2) The volume of a cube is given by the cube of the length of its side. In other words, if the length of each side is $x$, then the volume is $x^3$.

3) We are given that the volume of the chair is 343 cubic feet. Therefore, we can write an equation:

   $x^3 = 343$

4) To solve for $x$, we need to find the cube root of 343. The cube root of a number is a value that, when cubed (raised to the power of 3), gives the number.

5) The cube root of 343 is 7, because $7^3 = 7 \times 7 \times 7 = 343$.

Therefore, the length of each side of the chair is 7 feet.

### Answer: 7 feet.

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 Opus:

Let's approach this step by step:

1) We know that the volume of a cube is calculated by multiplying the length of one side by itself three times. In other words, if the length of one side is $s$, then the volume $V$ is:

   $V = s \times s \times s = s^3$

2) In this problem, we are given the volume and asked to find the length of one side. So, we need to work backwards from the volume formula:

   $729 = s^3$

3) To solve for $s$, we need to find the cube root of both sides:

   $\sqrt[3]{729} = \sqrt[3]{s^3}$

4) The cube root and the power of 3 cancel out on the right side:

   $\sqrt[3]{729} = s$

5) Now, let's calculate the cube root of 729:

   $\sqrt[3]{729} = 9$

Therefore, the length of each side of the cube is 9 inches.

### Answer: 9 inches.

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 Opus:

Let's approach this step-by-step:

1) We know that the spaceship is a cube. In a cube, all edges have the same length. Let's call this length $x$.

2) The volume of a cube is given by the cube of the length of its edge. In other words, if the length of an edge is $x$, then the volume is $x^3$.

3) We are given that the volume of the spaceship is 8 cubic miles. This means:

   $x^3 = 8$

4) To find $x$, we need to find the cube root of 8. The cube root of a number is a value that, when cubed, gives the number.

5) The cube root of 8 is 2, because $2^3 = 2 \times 2 \times 2 = 8$.

Therefore, the length of each edge of the spaceship is 2 miles.

### Answer: 2 miles

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 Opus:

Let's approach this step by step:

1) We know that the storage unit is a cube. In a cube, all sides have equal length. Let's call the length of each side $x$.

2) The volume of a cube is given by the cube of the length of its side. So, if the volume is 343 cubic feet and the length of each side is $x$ feet, we can write:

   $x^3 = 343$

3) To find $x$, we need to find the cube root of 343. 

   $x = \sqrt[3]{343}$

4) The cube root of 343 is 7.

   $x = 7$

5) Therefore, the length of each side of the storage unit is 7 feet.

### Answer: 7 feet.

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 Opus:

Let's approach this step by step:

1) The volume of a cube is given by the formula:

   Volume = edge length^3

2) We are given that the volume is 8 cubic inches. Let's call the edge length $x$ inches. Then we can write:

   8 = $x^3$

3) To solve for $x$, we need to take the cube root of both sides:

   $\sqrt[3]{8} = \sqrt[3]{x^3}$

   $2 = x$

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 Opus:

Let's approach this step by step:

1) We know that the volume of a cube is given by the formula:

   Volume = side length × side length × side length
   
   Or more concisely, if we let 's' be the side length:

   Volume = s³

2) In this problem, we are given the volume and asked to find the side length. So we need to solve the equation:

   729 = s³

3) To solve this, we need to find the cube root of 729. The cube root of a number is a value that, when cubed (raised to the power of 3), gives the number.

4) The cube root of 729 is 9, because 9³ = 9 × 9 × 9 = 729.

Therefore, the side length of the cubic plot is 9 meters.

### Answer: 9 meters.

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 Opus:

Let's solve this step by step:

1) We know that the slide is a cuboid (a rectangular prism) and all its sides are equal. Let's denote the length of one side as $x$ feet.

2) The volume of a cuboid is given by the formula: $V = l * w * h$, where $V$ is the volume, $l$ is the length, $w$ is the width, and $h$ is the height.

3) Since all sides are equal, we can replace $l$, $w$, and $h$ with $x$. So the formula becomes:

   $V = x * x * x = x^3$

4) We are given that the volume is 1000 cubic feet. So:

   $1000 = x^3$

5) To find $x$, we need to find the cube root of 1000:

   $x = \sqrt[3]{1000} = 10$

Therefore, the length of one side of the slide is 10 feet.

### Answer: 10 feet.

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 Opus:

Let's approach this step by step:

1) We know that the volume of a cube is given by the formula:

   Volume = side length × side length × side length
   
   Or, in mathematical terms: V = s³, where V is the volume and s is the side length.

2) We are given that the volume of the cube is 512 cubic feet. Let's substitute this into our formula:

   512 = s³

3) To find the side length, we need to find the cube root of 512. The cube root of a number is a value that, when cubed (raised to the power of 3), gives the number.

4) The cube root of 512 is 8, because 8³ = 8 × 8 × 8 = 512.

Therefore, the length of each side of the play cube is 8 feet.

### Answer: 8 feet.

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 Opus:

Let's approach this step by step:

1) We know that the hat, when fully inflated, is a perfect cube. This means that all its edges are equal in length.

2) Let's denote the length of one edge as $x$ centimeters.

3) The volume of a cube is given by the cube of the length of its edge. In other words, if the length of an edge is $x$, the volume is $x^3$.

4) We are told that the volume of the fully inflated hat is 27 cubic centimeters. This means:

   $x^3 = 27$

5) To find $x$, we need to find the cube root of 27. The cube root of a number is a value that, when cubed (raised to the power of 3), gives the number.

6) The cube root of 27 is 3, because $3^3 = 3 \times 3 \times 3 = 27$.

Therefore, the length of one edge of the hat when it is fully inflated is 3 centimeters.

### Answer: 3 centimeters.

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 Opus:

Let's approach this step by step:

1) We know that the container is cube-shaped. In a cube, all sides are equal.

2) We also know that the container holds 27 grapes in total. 

3) In a cube, the volume is calculated by multiplying the length of one side three times (since all sides are equal). 

   In other words, if the length of one side is s, then the volume of the cube is s × s × s, or s^3.

4) Now, we are told that the volume of the cube (the total number of grapes it holds) is 27. 

   So, s^3 = 27

5) To find the length of one side (s), we need to find the cube root of 27.

   The cube root of 27 is 3, because 3 × 3 × 3 = 27

6) Therefore, the length of each side of the cube is 3.

7) Since each side of the cube is a square, and we know the length of each side is 3, we can say that each side of the cube holds 3 × 3 = 9 grapes.

### Answer: 9

Expected answer: 3

Correct? No

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 Opus:

Sure, let's solve this problem step by step. To find the length of one side of the cube container, we need to use the formula for the volume of a cube.

Given:
- The volume of the container is 216 cubic meters.
- The container is in the shape of a cube.

Step 1: Recall the formula for the volume of a cube.
Volume of a cube = side length³

Step 2: Plug in the given volume and solve for the side length.
216 = side length³
∛216 = side length
6 = side length

Therefore, the length of one side of the cube container is 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 Opus:

Let's approach this step by step:

1) We know that the volume of a cube is given by the formula:

   Volume = edge length^3

2) We are given that the volume is 64 mm^3. Let's call the edge length x. Then:

   64 = x^3

3) To find x, we need to take the cube root of both sides:

   ∛64 = x

4) The cube root of 64 is 4, so:

   4 = x

Therefore, the length of each edge of the cube charm is 4 millimeters.

### Answer: 4 millimeters.

Expected answer: 4

Correct? Yes

Problem: If a number is cubed and the result is 64, what is the number?

Claude-3 Opus:

Let's approach this step by step:

1) We are told that if a number is cubed, the result is 64. In mathematical terms, this can be written as:

   $x^3 = 64$

   where $x$ is the unknown number.

2) To find $x$, we need to find the cube root of 64. The cube root of a number is a value that, when cubed, gives the number.

3) The cube root of 64 can be written as:

   $\sqrt[3]{64} = 4$

   This is because $4^3 = 4 \times 4 \times 4 = 64$.

4) Therefore, the number we are looking for 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 Opus:

Okay, let's think through this step-by-step:

1) We know that the mug is cubic in shape, which means all sides are equal in length. Let's call the length of each side $x$.

2) The volume of a cube is given by the formula: $V = x^3$, where $V$ is the volume and $x$ is the length of each side.

3) We are told that the mug accommodates 1 liter of hot drink. Therefore, the volume of the mug is 1 liter. 

4) Now, we can substitute this into our formula:

   1 liter = $x^3$

5) To find $x$, we need to take the cube root of both sides:

   $\sqrt[3]{1 \text{ liter}} = x$

6) The cube root of 1 is 1. Therefore:

   $x = 1$ liter

So, each side of the petite mug is 1 liter long.

### Answer: 1 liter

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 Opus:

Let's approach this step by step:

1) We know that the volume of a cube is given by the formula:

   Volume = side length³

2) In this case, the volume is 343 cubic inches. So we can write:

   343 = side length³

3) To find the side length, we need to find the cube root of 343. 

4) The cube root of 343 is 7, because 7³ = 7 * 7 * 7 = 343.

5) Therefore, the side length of the cube is 7 inches.

### Answer: 7 inches.

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 Opus:

Let's approach this step by step:

1) We know that the volume of the larger cube is 729 cubic cm. The volume of a cube is given by the cube of its side length. So, if we call the side length of the larger cube $L$, we can write:

   $L^3 = 729$

2) To find $L$, we need to find the cube root of 729:

   $L = \sqrt[3]{729} = 9$

   So, the side length of the larger cube is 9 cm.

3) Now, the larger cube is made up of smaller cubes. If each side of the smaller cube is $l$ cm, then the number of smaller cubes along each side of the larger cube is $9/l$.

4) Since the larger cube is a perfect cube, the number of smaller cubes along each side must be an integer. This means $l$ must be a factor of 9.

5) The factors of 9 are 1, 3, and 9. So, the possible values of $l$ are 1 cm, 3 cm, and 9 cm.

6) However, the question states that the larger cube is made up of more than one smaller cube. So, $l$ cannot be 9 cm (as that would mean the larger cube is made up of only one smaller cube).

7) Therefore, the side length of each smaller cube is either 1 cm or 3 cm.

### Answer: The side length of each smaller ice cube is either 1 cm or 3 cm.

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 Opus:

Let's approach this step by step:

1) We know that the volume of a cube is given by the formula:

   Volume = side length³

2) In this case, the volume is 216 cubic cm. So we can write:

   216 = side length³

3) To find the side length, we need to find the cube root of 216. 

4) The cube root of 216 is 6, because 6³ = 6 × 6 × 6 = 216.

Therefore, the length of each side of the wheel is 6 cm.

### Answer: 6 cm.

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 Opus:

Let's approach this step by step:

1) We know that the flower bud is a cube. In a cube, all edges are equal in length. Let's call the length of each edge $x$ millimeters.

2) The volume of a cube is given by the cube of the length of its edge. In other words, if the length of each edge is $x$, then the volume is $x^3$.

3) We are told that the volume of the bud is 216 cubic millimeters. Therefore, we can set up an equation:

   $x^3 = 216$

4) To solve for $x$, we need to find the cube root of 216. The cube root of 216 is 6, because $6 \times 6 \times 6 = 216$.

   $x = \sqrt[3]{216} = 6$

5) Therefore, the length of each edge of the flower bud is 6 millimeters.

### Answer: 6 millimeters.

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 Opus:

Let's approach this step by step:

1) We know that the shell is in the shape of a cube. In a cube, all sides are equal in length. Let's call the length of each side $x$ cm.

2) The volume of a cube is given by the cube of the length of its side. In other words, if the length of each side is $x$ cm, then the volume is $x^3$ cm^3.

3) We are given that the volume of the shell is 27 cm^3. Therefore, we can write the equation:

   $x^3 = 27$

4) To solve for $x$, we need to find the cube root of both sides:

   $\sqrt[3]{x^3} = \sqrt[3]{27}$
   
   $x = 3$

Therefore, the length of each side of the snail's cubical shell is 3 cm.

### Answer: 3 cm

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 Opus:

Let's approach this step by step:

1) We know that the balloon is a cube. In a cube, all sides are equal in length. Let's call the length of one side $s$.

2) The volume of a cube is given by the formula:

   Volume = $s^3$

3) We are told that the volume of the balloon is 27 cubic feet. So we can write:

   27 = $s^3$

4) To find $s$, we need to take the cube root of both sides:

   $\sqrt[3]{27} = \sqrt[3]{s^3}$

5) The cube root and the power of 3 cancel out on the right side:

   $\sqrt[3]{27} = s$

6) Now, let's calculate the cube root of 27:

   $\sqrt[3]{27} = 3$

So, the length of one side of the balloon is 3 feet.

### Answer: 3 feet.

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 Opus:

Let's solve this step by step:

1) We know that the tank is a cube. In a cube, all edges have the same length. Let's call this length $x$.

2) The volume of a cube is given by the cube of the length of its edge. In other words, if the length of an edge is $x$, the volume is $x^3$.

3) We are told that the volume of the tank is 216 cubic feet. So we can write an equation:

   $x^3 = 216$

4) To solve for $x$, we need to find the cube root of both sides:

   $\sqrt[3]{x^3} = \sqrt[3]{216}$
   
   $x = 6$

Therefore, the length of one edge of the tank is 6 feet.

### Answer: 6 feet.

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 Opus:

Let's approach this step by step:

1) We know that the box is a cube, which means all sides are equal in length. Let's call the length of one side $x$ inches.

2) The volume of a cube is given by the cube of the length of its side. In other words, if the length of each side is $x$ inches, then the volume is $x^3$ cubic inches.

3) We are given that the volume of the box is 216 cubic inches. Therefore, we can write the equation:

   $x^3 = 216$

4) To solve for $x$, we need to find the cube root of 216. 

   $x = \sqrt[3]{216}$

5) The cube root of 216 is 6.

   $x = 6$

Therefore, the length of one side of the box is 6 inches.

### Answer: 6 inches.

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 Opus:

Let's approach this step by step:

1) We know that the pocket is in the shape of a cube. In a cube, all edges have the same length. Let's call this length $x$.

2) The volume of a cube is given by the cube of the length of its edge. In other words, if the length of each edge is $x$, then the volume is $x^3$.

3) We are told that the volume of the pocket is 125 cubic inches. Therefore, we can write an equation:

   $x^3 = 125$

4) To solve for $x$, we need to find the cube root of both sides:

   $\sqrt[3]{x^3} = \sqrt[3]{125}$
   
   $x = 5$

5) Therefore, the length of each edge of the 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 Opus:

Let's solve this step by step:

1) We know that the flowers are arranged in a cube shape. In a cube, all sides have equal length. Let's call the length of each side $x$.

2) In a cube, the volume is calculated by cubing the length of a side. So, the volume of the flower cube is $x^3$.

3) We also know that the total number of flowers used is 125. This means that the volume of the cube (in terms of number of flowers) is 125.

4) We can write an equation:

   $x^3 = 125$

5) To find $x$, we need to find the cube root of 125:

   $x = \sqrt[3]{125} = 5$

Therefore, the length of each side of the flower cube is 5.

### Answer: 5

Expected answer: 5

Correct? Yes

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 Opus:

Let's approach this step by step:

1) We know that the cucumber is in the shape of a perfect cube. In a cube, all sides have equal length. Let's call the length of each side $x$ centimeters.

2) The volume of a cube is given by the cube of the length of its side. So, if the volume is 1000 cubic centimeters and the length of each side is $x$ centimeters, we can write:

   $x^3 = 1000$

3) To find $x$, we need to find the cube root of 1000. 

   $x = \sqrt[3]{1000}$

4) The cube root of 1000 is 10.

   $x = 10$

Therefore, the length of each side of the cubic cucumber is 10 centimeters.

### Answer: 10 centimeters.

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 Opus:

Let's approach this step by step:

1) We know that the volume of the cargo hold is 27 cubic meters. Since all sides of the hold have equal length, the cargo hold is in the shape of a cube.

2) The volume of a cube is given by the formula: 
   
   Volume = length^3

   where length is the length of one side of the cube.

3) In this case, we have:

   27 = length^3

4) To find the length, we need to find the cube root of 27:

   length = ∛27

5) The cube root of 27 is 3, because 3 * 3 * 3 = 27.

   length = 3

Therefore, the length of one side of the cargo hold is 3 meters.

### Answer: 3 meters.

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 Opus:

Let's approach this step by step:

1) We know that the cucumber is in the shape of a perfect cube. In a cube, all sides have equal length. Let's call the length of each side $x$ centimeters.

2) The volume of a cube is given by the cube of the length of its side. In other words, if the length of each side is $x$, then the volume is $x^3$.

3) We are told that the volume of the cucumber is 216 cubic centimeters. Therefore, we can write the equation:

   $x^3 = 216$

4) To find $x$, we need to find the cube root of 216. The cube root of a number is a value that, when cubed (raised to the power of 3), gives the number.

5) The cube root of 216 is 6, because $6^3 = 6 \times 6 \times 6 = 216$.

Therefore, the length of each side of the cubic cucumber is 6 centimeters.

### Answer: 6 centimeters.

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 Opus:

Let's approach this step by step:

1) We know that the box is a cube. In a cube, all edges have the same length. Let's call this length $x$.

2) The volume of a cube is given by the cube of its edge length. So, if the volume is 729 cubic feet, we can write:

   $x^3 = 729$

3) To find $x$, we need to find the cube root of 729. 

   $x = \sqrt[3]{729}$

4) The cube root of 729 is 9.

   $x = 9$

Therefore, the length of one edge of the box is 9 feet.

### Answer: 9 feet.

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 Opus:

Let's approach this step by step:

1) We know that the number 27 is obtained by cubing a letter's position in the alphabet. In other words, if the letter's position is n, then:

   n^3 = 27

2) To find n, we need to find the cube root of 27:

   n = ∛27

3) The cube root of 27 is 3:

   n = 3

4) Now, we need to find the letter that corresponds to the position 3 in the alphabet. We're using 'a' as 1, 'b' as 2, and so on. So, the letter in the 3rd position is 'c'.

Therefore, the letter corresponding to the number 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 Opus:

Let's approach this step by step:

1) We know that the buoy is a cube. In a cube, all sides have equal length. Let's call the length of each side $x$ feet.

2) The volume of a cube is given by the cube of the length of its side. In other words, if the length of each side is $x$, then the volume is $x^3$.

3) We are given that the volume of the buoy is 1 cubic foot. Therefore, we can write an equation:

   $x^3 = 1$

4) To solve for $x$, we need to find the cube root of both sides:

   $\sqrt[3]{x^3} = \sqrt[3]{1}$

   $x = 1$

5) Therefore, the length of each side of the buoy is 1 foot.

### Answer: 1 foot

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 Opus:

Let's approach this step by step:

1) Let the length of the object be $l$ units and the constant of proportionality be $k$.

2) Given that the temperature at the heart of the object is directly proportional to the cube of its length, we can write:

   $T = kl^3$

   where $T$ is the temperature.

3) We are told that the temperature is 729 degrees. Substituting this into the equation:

   $729 = kl^3$

4) Now, we need to find the value of $k$. In the metric system, the unit of temperature is Kelvin (K) and the unit of length is meter (m). In the SI system, the unit of temperature is degree Celsius (°C) and the unit of length is also meter (m).

   In both systems, when the length is 1 unit (i.e., 1 m), the temperature would also be 1 unit (1 K or 1 °C). This means:

   $1 = k(1)^3$

   Therefore, $k = 1$.

5) Substituting $k = 1$ into the equation from step 3:

   $729 = 1 \cdot l^3$

   $l^3 = 729$

   $l = \sqrt[3]{729} = 9$

Therefore, the length of the object is 9 units.

### Answer: 9 units.

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 Opus:

Let's solve this step by step:

1) We know that the volume of a cube is given by the formula:
   
   Volume = side length³

2) In this case, the volume is 8 cubic centimeters. So we can write:
   
   8 = side length³

3) To find the side length, we need to find the cube root of both sides:
   
   ∛8 = ∛(side length³)
   
   ∛8 = side length

4) The cube root of 8 is 2, because 2³ = 8.

   2 = side length

Therefore, the length of each side of the cube-shaped orange is 2 centimeters.

### Answer: 2 centimeters.

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 Opus:

Let's approach this step by step:

1) We know that the volume of a cube is given by the formula:

   Volume = edge length³

2) In this case, the volume is 729 cubic centimeters. So we can write:

   729 = edge length³

3) To find the edge length, we need to find the cube root of 729. In other words, what number multiplied by itself three times gives 729?

4) The cube root of 729 is 9, because 9 * 9 * 9 = 729.

5) Therefore, the edge length of the cubical mirror box is 9 centimeters.

### Answer: 9 centimeters.

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 Opus:

Let's approach this step by step:

1) We know that the storage compartment is a cube. In a cube, all sides have equal length. Let's call the length of one side $x$ inches.

2) The volume of a cube is given by the cube of the length of its side. In other words, if the length of each side is $x$, then the volume is $x^3$.

3) We are given that the volume of the compartment is 125 cubic inches. Therefore, we can write the equation:

   $x^3 = 125$

4) To find $x$, we need to find the cube root of 125. The cube root of a number is a value that, when cubed, gives the number. 

5) The cube root of 125 is 5, because $5^3 = 5 \times 5 \times 5 = 125$.

Therefore, the length of each side of the cubical storage compartment is 5 inches.

### Answer: 5 inches.

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 Opus:

Let's approach this step-by-step:

1) We know that the box is a cube. In a cube, all sides have equal length. Let's call the length of each side $x$ inches.

2) The volume of a cube is given by the cube of the length of its side. In other words, if the length of each side is $x$, then the volume is $x^3$.

3) We are given that the volume of the box is 343 cubic inches. Therefore, we can write the equation:

   $x^3 = 343$

4) To find $x$, we need to find the cube root of 343. The cube root of a number is a value that, when cubed, gives the number.

5) The cube root of 343 is 7, because $7^3 = 7 \times 7 \times 7 = 343$.

Therefore, the length of each side of the box is 7 inches.

### Answer: 7 inches.

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 Opus:

Let's approach this step by step:

1) We know that the package is cube-shaped. In a cube, all edges are equal in length. Let's call the length of each edge $x$ centimeters.

2) The volume of a cube is given by the cube of the length of its edge. In other words, if the length of each edge is $x$, then the volume is $x^3$.

3) We are told that the volume of the package is 512 cubic centimeters. This means:

   $x^3 = 512$

4) To find $x$, we need to find the cube root of 512. The cube root of a number is a value that, when cubed (raised to the power of 3), gives the number.

5) The cube root of 512 is 8, because $8^3 = 8 \times 8 \times 8 = 512$.

Therefore, the length of each edge of the candy package is 8 centimeters.

### Answer: 8 centimeters.

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 Opus:

Let's approach this step by step:

1) We know that the volume of a cube is given by the formula:

   Volume = side length^3

2) In this case, the volume is 512 cubic feet. So we can write:

   512 = side length^3

3) To find the side length, we need to find the cube root of 512. 

4) The cube root of 512 is 8, because 8 * 8 * 8 = 512.

Therefore, the length of each side of the jungle gym is 8 feet.

### Answer: 8 feet.

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 Opus:

Let's approach this step by step:

1) We know that the toothbrush is a perfect cube. In a cube, all edges have the same length. Let's call this length $x$ centimeters.

2) The volume of a cube is given by the cube of the length of its edge. In other words, if the length of each edge is $x$, then the volume is $x^3$.

3) We are told that the volume of the toothbrush is 729 cubic centimeters. Therefore, we can write an equation:

   $x^3 = 729$

4) To solve for $x$, we need to find the cube root of 729. The cube root of a number is a value that, when cubed (raised to the power of 3), gives the number.

5) The cube root of 729 is 9, because $9^3 = 9 \times 9 \times 9 = 729$.

Therefore, the length of each edge of the toothbrush is 9 centimeters.

### Answer: 9 centimeters.

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 Opus:

Let's approach this step by step:

1) We know that the star is a perfect cube. In a cube, all edges have the same length. Let's call this length $x$ light years.

2) The volume of a cube is given by the cube of its edge length. So if the edge length is $x$, the volume is $x^3$.

3) We are told that the volume of the star is 8 cubic light years. This means:

   $x^3 = 8$

4) To find $x$, we need to take the cube root of both sides:

   $\sqrt[3]{x^3} = \sqrt[3]{8}$
   
   $x = 2$

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 Opus:

Let's approach this step by step:

1) We know that the dolphin's call can fill a cubical chamber. This means that the volume of the call is equal to the volume of the chamber.

2) The volume of the call is given as 1 unit^3. This means that the volume of the chamber is also 1 unit^3.

3) For a cube, the volume is equal to the cube of the length of its side. If we denote the length of each side as 's', then:

   Volume = s^3

4) We know the volume is 1 unit^3, so:

   1 = s^3

5) To find 's', we need to find the cube root of both sides:

   ∛1 = ∛(s^3)
   1 = s

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 Opus:

Let's approach this step by step:

1) The volume of a cube is given by the formula:

   Volume = side length^3

2) We are given that the volume is 1000 mm^3. Let's call the side length s. Then:

   1000 = s^3

3) To find s, we need to take the cube root of both sides:

   ∛1000 = ∛(s^3)
   
   ∛1000 = s

4) The cube root of 1000 is 10.

Therefore, the length of each side of the cubic gadget is 10 millimeters.

### Answer: 10 millimeters.

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 Opus:

To find the length of one edge of the cube, we need to use the formula for the volume of a cube:

Volume = edge length³

Given:
- The volume of the cube is 1 cubic meter.

Let's solve for the edge length:

1 = edge length³
∛1 = edge length (taking the cube root of both sides)
1 = edge length (since the cube root of 1 is 1)

Therefore, the length of one edge of the cube is 1 meter.

### Answer: 1 meter.

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 Opus:

Let's approach this step-by-step:

1) We know that the box is a cube. In a cube, all sides are equal in length. Let's call the length of each side $s$.

2) The volume of a cube is given by the cube of the length of its side. In other words, if the length of each side is $s$, then the volume is $s^3$.

3) We are given that the volume of the box is 216 cubic meters. So we can write:

   $s^3 = 216$

4) To find $s$, we need to find the cube root of 216. The cube root of a number is a value that, when cubed, gives the number. 

5) The cube root of 216 is 6, because $6^3 = 6 \times 6 \times 6 = 216$.

Therefore, the length of each side of the magical cube-shaped box is 6 meters.

### Answer: 6 meters

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 Opus:

Let's approach this step by step:

1) We know that the house is cube-shaped. In a cube, all sides are equal in length. Let's call the length of one side $s$.

2) The volume of a cube is given by the cube of the length of its side. In other words, if the length of each side is $s$, then the volume is $s^3$.

3) We are given that the volume of the house is 64 cubic feet. Therefore, we can write an equation:

   $s^3 = 64$

4) To find $s$, we need to find the cube root of 64. The cube root of a number is a value that, when cubed, gives the number.

5) The cube root of 64 is 4, because $4^3 = 4 \times 4 \times 4 = 64$.

Therefore, the length of one side of the house is 4 feet.

### Answer: 4 feet.

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 Opus:

Let's approach this step by step:

1) We know that the volume of a cube is given by the formula:

   Volume = edge length^3

2) We are given that the volume is 1000 cubic centimeters. Let's call the edge length x centimeters. Then we can write:

   1000 = x^3

3) To solve for x, we need to take the cube root of both sides:

   ∛1000 = ∛(x^3)

   ∛1000 = x

4) The cube root of 1000 is 10. So:

   x = 10

Therefore, the length of the edge of the toothbrush holder is 10 centimeters.

### Answer: 10 centimeters.

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 Opus:

To solve this problem, let's use the formula for the volume of a cube:

Volume of a cube = side length³

Given:
- The bread loaf is in a cubic form.
- The volume of the cube-shaped oven is 125 cubic inches.

Step 1: Let the side length of the bread loaf be f inches.

Step 2: Apply the volume formula.
Volume = f³
125 = f³

Step 3: Solve for f by taking the cube root of both sides.
f³ = 125
f = ∛125
f = 5

Therefore, the length of each side of the bread loaf is 5 inches.

### Answer: 5 inches.

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 Opus:

Let's approach this step by step:

1) We know that the strawberry is in the shape of a cube. In a cube, all sides are equal in length. Let's call the length of one side of the cube $x$ millimeters.

2) The volume of a cube is given by the cube of the length of its side. In other words, if the length of a side is $x$, the volume is $x^3$.

3) We are told that the volume of the strawberry is 64 cubic millimeters. Therefore, we can write an equation:

   $x^3 = 64$

4) To solve for $x$, we need to find the cube root of both sides:

   $\sqrt[3]{x^3} = \sqrt[3]{64}$
   
   $x = 4$

Therefore, the length of one side of the strawberry is 4 millimeters.

### Answer: 4 millimeters.

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 Opus:

Let's solve this step by step:

1) We know that the rocket is a perfect cube. In a cube, all sides have equal length. Let's call the length of each side $x$ meters.

2) The volume of a cube is given by the cube of its side length. So, if the volume is 216 cubic meters and the side length is $x$ meters, we can write:

   $x^3 = 216$

3) To find $x$, we need to find the cube root of 216. 

   $x = \sqrt[3]{216}$

4) The cube root of 216 is 6.

   $x = 6$

5) Therefore, the length of each side of the rocket is 6 meters.

### Answer: 6 meters.

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 Opus:

Let's solve this step by step:

1) We know that the lamp base is a cube. In a cube, all edges have the same length. Let's call this length $x$.

2) The volume of a cube is given by the cube of the length of its edge. In other words, if the length of an edge is $x$, the volume is $x^3$.

3) We are given that the volume of the cube is 512 cubic inches. So we can write an equation:

   $x^3 = 512$

4) To solve for $x$, we need to find the cube root of both sides:

   $\sqrt[3]{x^3} = \sqrt[3]{512}$

   $x = 8$

5) Therefore, the length of one edge of the 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 Opus:

Let's approach this step by step:

1) We know that the eggs are arranged in a cubical shape. This means that the number of eggs is equal to the cube of the length of the edge.

2) Let's denote the length of the edge as $x$. Then, the number of eggs is $x^3$.

3) We are told that the number of eggs is 27. So we can write an equation:

   $x^3 = 27$

4) To solve this, we need to find the cube root of 27. The cube root of a number is a value that, when cubed (raised to the power of 3), gives the number.

5) The cube root of 27 is 3, because $3^3 = 3 \times 3 \times 3 = 27$.

Therefore, the length of the edge of the cubical arrangement 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 Opus:

Let's approach this step by step:

1) We know that the volume of the bus is 512 cubic meters. Let's call the length of one edge of the cube $x$ meters.

2) The volume of a cube is given by the cube of its edge length. So, we can write an equation:

   $x^3 = 512$

3) To solve for $x$, we need to find the cube root of both sides:

   $\sqrt[3]{x^3} = \sqrt[3]{512}$

   $x = \sqrt[3]{512}$

4) The cube root of 512 is 8.

   $x = 8$

Therefore, the length of one edge of the cube approximating the bus is 8 meters.

### Answer: 8 meters.

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 Opus:

Let's approach this step by step:

1) We know that the volume of a cube is given by the formula:

   Volume = edge length³

2) In this case, the volume is 343 cubic meters. So we can write:

   343 = edge length³

3) To find the edge length, we need to find the cube root of 343. 

4) The cube root of 343 is 7, because 7³ = 7 * 7 * 7 = 343.

5) Therefore, the edge length of the balloon is 7 meters.

### Answer: 7 meters.

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 Opus:

Let's approach this step by step:

1) We know that the shirt design is in the shape of a cube. A cube has all sides of equal length.

2) Let's denote the length of one side of the cube as 'l'. 

3) The volume of a cube is given by the formula: 
   Volume = l³, where l is the length of one side of the cube.

4) We are given that the volume of the shirt design is 343 cubic centimeters. So,
   343 = l³

5) To find the length of one side, we need to find the cube root of 343.
   l = ³√343
   
6) The cube root of 343 is 7, because 7 * 7 * 7 = 343.
   l = 7

Therefore, the length of one side of this cubic shirt design is 7 centimeters.

### Answer: 7 centimeters.

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 Opus:

Let's approach this step by step:

1) We know that the roller skate container is cube-shaped. In a cube, all sides have equal length. Let's call the length of each side $x$ inches.

2) The volume of a cube is given by the cube of the length of its side. In other words, if the length of each side is $x$ inches, the volume is $x^3$ cubic inches.

3) We are given that the volume of the container is 216 cubic inches. Therefore, we can write the equation:

   $x^3 = 216$

4) To find $x$, we need to find the cube root of 216. The cube root of 216 is 6 because $6 \times 6 \times 6 = 216$.

   $x = \sqrt[3]{216} = 6$

Therefore, the length of each side of the roller skate container is 6 inches.

### Answer: 6 inches.

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 Opus:

Let's solve this step by step:

1) We know that the volume of a cube is given by the formula:

   Volume = side length³

2) In this case, the volume is 343 cubic meters. So we can write:

   343 = side length³

3) To find the side length, we need to find the cube root of 343. 

4) The cube root of 343 is 7, because 7³ = 7 × 7 × 7 = 343.

Therefore, the side length of the cubic water tank is 7 meters.

### Answer: 7 meters.

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 Opus:

Let's approach this step by step:

1) We know that the backpack is a cube. In a cube, all sides have equal length. Let's call the length of each side $x$.

2) The volume of a cube is given by the cube of the length of its side. In other words, if the length of each side is $x$, then the volume is $x^3$.

3) We are given that the volume of the backpack is 216 cubic inches. Therefore, we can write the equation:

   $x^3 = 216$

4) To find $x$, we need to find the cube root of 216. The cube root of a number is a value that, when cubed, gives the number.

5) The cube root of 216 is 6, because $6^3 = 6 \times 6 \times 6 = 216$.

Therefore, the length of each side of the backpack is 6 inches.

### Answer: 6 inches

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 Opus:

Let's approach this step-by-step:

1) We know that the dinosaur's brain is cube-shaped. In a cube, all sides have equal length. Let's call this length $l$.

2) The volume of a cube is given by the formula:

   Volume = $l^3$, where $l$ is the length of each side.

3) We are given that the volume of the dinosaur's brain is 1 cubic meter. So:

   1 = $l^3$

4) To find $l$, we need to take the cube root of both sides:

   $\sqrt[3]{1} = \sqrt[3]{l^3}$

5) The cube root and the power of 3 cancel out on the right side:

   $\sqrt[3]{1} = l$

6) The cube root of 1 is 1:

   1 = $l$

Therefore, the length of each side of the dinosaur's brain's cube structure is 1 meter.

### Answer: 1 meter.

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 Opus:

Let's approach this step by step:

1) We know that the volume of a cube is given by the formula:

   Volume = edge length^3

2) In this case, the volume is 1000 cubic centimeters. Let's call the edge length x centimeters. So we can write:

   1000 = x^3

3) To find x, we need to take the cube root of both sides:

   ∛1000 = ∛(x^3)

   ∛1000 = x

4) The cube root of 1000 is 10.

Therefore, the edge length of the flower cube is 10 centimeters.

### Answer: 10 centimeters.

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 Opus:

Let's approach this step by step:

1) We know that the box is cube-shaped. This means that all sides of the box are equal in length.

2) Let's denote the length of one side of the box as 'x' inches.

3) The volume of a cube is given by the cube of the length of its side. In other words, if the length of each side is 'x', then the volume is 'x³'.

4) We are told that the volume of the box is 512 cubic inches. Therefore:

   x³ = 512

5) To find the value of 'x', we need to find the cube root of 512.

   x = ∛512 = 8

Therefore, the length of one side of the magical cube-shaped box is 8 inches.

### Answer: 8 inches.

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 Opus:

Let's approach this step by step:

1) We know that the toy car is in the shape of a cube. In a cube, all sides have equal length. Let's call the length of each side $x$ inches.

2) The volume of a cube is given by the cube of the length of its side. In other words, if the length of each side is $x$, then the volume is $x^3$.

3) We are given that the volume of the toy car is 64 cubic inches. Therefore, we can write the equation:

   $x^3 = 64$

4) To solve for $x$, we need to find the cube root of 64. The cube root of 64 is 4, because $4 \times 4 \times 4 = 64$.

   $x = \sqrt[3]{64} = 4$

5) 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 Opus:

Let's solve this step by step:

1) We know that the clock is cube-shaped. In a cube, all sides have equal length. Let's call the length of each side $x$ inches.

2) The volume of a cube is given by the cube of the length of its side. So, if the volume is 27 cubic inches, we can write:

   $x^3 = 27$

3) To find $x$, we need to take the cube root of both sides:

   $\sqrt[3]{x^3} = \sqrt[3]{27}$
   
   $x = 3$

Therefore, the length of each side of the clock is 3 inches.

### Answer: 3 inches.

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 Opus:

Let's solve this step by step:

1) Let's denote the length of a side (which is the same as the height and width) as $x$ meters.

2) The volume of a cube is given by the cube of its side length. So, if the volume is 8 cubic meters, we can write:

   $x^3 = 8$

3) To find $x$, we need to find the cube root of both sides:

   $\sqrt[3]{x^3} = \sqrt[3]{8}$

   $x = 2$

4) Therefore, the length of a side of the cubical cargo space is 2 meters.

### Answer: 2 meters.

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 Opus:

Let's approach this step by step:

1) We know that the box is a cube, which means all sides are equal in length.

2) We also know that the box perfectly fits 343 grapes. This means that the number of grapes along each side of the box, when multiplied together, equals 343.

3) In mathematical terms, if we let the side length be x grapes, then:

   x * x * x = 343

   or 

   x^3 = 343

4) To find x, we need to find the cube root of 343.

5) The cube root of 343 is 7, because 7 * 7 * 7 = 343.

Therefore, the length of each side of the box is 7 grapes.

### Answer: 7 grapes

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 Opus:

Let's approach this step by step:

1) We know that the volume of a cube is given by the formula:

   Volume = side length^3

2) We are given that the volume of the elephant is 343 cubic meters. Let's call the side length 's'. Then:

   343 = s^3

3) To find the side length, we need to find the cube root of 343:

   s = ∛343

4) The cube root of 343 is 7, because 7 * 7 * 7 = 343.

   s = 7

5) Therefore, if an elephant with a volume of 343 cubic meters had a perfect cube shape, each side of the cube would be 7 meters long.

### Answer: 7 meters.

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 Opus:

Let's solve this step by step:

1) We know that the box is a cube. In a cube, all edges have the same length. Let's call the length of an edge $x$ centimeters.

2) The volume of a cube is given by the cube of the length of its edge. So, if the edge length is $x$ cm, the volume is $x^3$ cubic centimeters.

3) We are told that the volume of each box is 27 cubic centimeters. So we can write an equation:

   $x^3 = 27$

4) To solve this, we need to find the cube root of 27. The cube root of a number is a value that, when cubed (raised to the power of 3), gives the number.

5) The cube root of 27 is 3, because $3^3 = 3 \times 3 \times 3 = 27$.

Therefore, the length of an edge of the box is 3 centimeters.

### Answer: 3 centimeters.

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 Opus:

Let's approach this step by step:

1) We know that the territory is cube-shaped. In a cube, the volume is equal to the cube of the edge length. If we denote the edge length as $l$, then the volume $V$ is:

   $V = l^3$

2) We are given that the volume of the territory is 8 km^3. Let's substitute this into the formula:

   $8 = l^3$

3) To find the edge length, we need to find the cube root of both sides:

   $\sqrt[3]{8} = \sqrt[3]{l^3}$

4) The cube root and the power of 3 cancel out on the right side:

   $\sqrt[3]{8} = l$

5) Now, let's calculate the cube root of 8:

   $\sqrt[3]{8} = 2$

Therefore, the edge length of the lion's territory is 2 km.

### Answer: 2 km

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 Opus:

Let's solve this step by step:

1) We know that the backpack is a cube, and the volume of a cube is given by the formula:

   Volume = side length × side length × side length
   
   or
   
   Volume = side length^3

2) We are given that the volume of the backpack is 125 cubic inches. Let's call the side length of the cube 's'. Then:

   125 = s^3

3) To find 's', we need to find the cube root of 125. The cube root of a number is a value that, when cubed (raised to the power of 3), gives the number.

4) The cube root of 125 is 5, because 5^3 = 5 × 5 × 5 = 125.

   So, s = 5.

5) 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 Opus:

Let's solve this step by step:

1) We know that the volume of a cube is given by the formula:

   Volume = side length³

2) In this case, the volume is 64 cubic meters. So we can write:

   64 = side length³

3) To find the side length, we need to find the cube root of 64. The cube root of a number is a value that, when cubed (raised to the power of 3), gives the number.

4) The cube root of 64 is 4, because 4³ = 4 × 4 × 4 = 64.

5) Therefore, the side length of the cube is 4 meters.

### Answer: 4 meters.

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 Opus:

Let's approach this step by step:

1) We know that the treasure chest is a cube. In a cube, all sides have equal length. Let's call the length of each side $s$.

2) The volume of a cube is given by the cube of the length of its side. In other words, if the length of each side is $s$, then the volume is $s^3$.

3) We are given that the volume of the treasure chest is 125 cubic feet. Therefore, we can write an equation:

   $s^3 = 125$

4) To find $s$, we need to find the cube root of 125. The cube root of a number is a value that, when cubed, gives the number.

5) The cube root of 125 is 5, because $5^3 = 5 \times 5 \times 5 = 125$.

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 Opus:

Let's approach this step by step:

1) We know that the fish tank is a cube. In a cube, all sides have equal length. Let's call the length of each side $x$ meters.

2) The volume of a cube is given by the cube of the length of its side. In other words, if the length of each side is $x$, then the volume is $x^3$.

3) We are told that the volume of the fish tank is exactly 1 cubic meter. This means:

   $x^3 = 1$

4) To find $x$, we need to find the cube root of 1. The cube root of a number is a value that, when cubed, gives the number. 

5) The cube root of 1 is 1, because $1^3 = 1 \times 1 \times 1 = 1$.

Therefore, $x = 1$.

### Answer: 1 meter

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 Opus:

Let's approach this step by step:

1) We know that the volume of a cube is given by the formula:

   Volume = side length³

2) In this case, the volume is 125 cubic centimeters. Let's call the side length $s$ centimeters. So we have:

   125 = $s$³

3) To find $s$, we need to take the cube root of both sides:

   $\sqrt[3]{125}$ = $\sqrt[3]{s^3}$

4) The cube root and the power of 3 cancel out on the right side:

   $\sqrt[3]{125}$ = $s$

5) Now, let's calculate the cube root of 125:

   $\sqrt[3]{125}$ = 5

Therefore, the side length of the cheese cube is 5 centimeters.

### Answer: 5 centimeters.

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 Opus:

Let's approach this step by step:

1) We know that the skatepark is a perfect cube. In a cube, all edges have the same length. Let's call this length $x$ meters.

2) The volume of a cube is given by the cube of its edge length. So, if the edge length is $x$, the volume is $x^3$.

3) We are told that the volume of the skatepark is 343 cubic meters. This means:

   $x^3 = 343$

4) To find $x$, we need to find the cube root of 343. The cube root of a number is a value that, when cubed, gives the number.

5) The cube root of 343 is 7, because $7^3 = 7 \times 7 \times 7 = 343$.

Therefore, the length of each edge of the skatepark is 7 meters.

### Answer: 7 meters.

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 Opus:

Let's approach this step by step:

1) We know that the box is a cube, which means all sides have equal length. Let's call the length of each side $x$.

2) The volume of a cube is given by the cube of the length of its side. In this case, we are told that the volume is 1 cubic foot. So we can write:

   $x^3 = 1$

3) To find $x$, we need to take the cube root of both sides:

   $\sqrt[3]{x^3} = \sqrt[3]{1}$

   $x = 1$

Therefore, the length of each side of the box is 1 foot.

### Answer: 1 foot

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 Opus:

Let's approach this step-by-step:

1) We know that the volume of a cube is given by the formula:

   Volume = side length × side length × side length
   
   Or more concisely, if we let 's' be the side length:

   Volume = s³

2) We are given that the volume is 729 cubic units. So we can set up an equation:

   729 = s³

3) To solve for 's', we need to find the cube root of both sides:

   ∛729 = ∛(s³)

   The cube root and the cube cancel out on the right side, leaving us with:

   ∛729 = s

4) The cube root of 729 is 9. So:

   9 = s

Therefore, the length of each side of the item cube is 9 units.

### Answer: 9 units.

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 Opus:

Let's approach this step by step:

1) We know that the cookie dough is in the shape of a cube. In a cube, all sides have equal length. Let's call the length of each side $x$ cm.

2) The volume of a cube is given by the cube of its side length. So, if the volume is 512 cubic cm and the side length is $x$ cm, we can write:

   $x^3 = 512$

3) To find $x$, we need to find the cube root of 512. 

   $x = \sqrt[3]{512}$

4) The cube root of 512 is 8.

   $x = 8$

5) Therefore, the length of each side of the cube is 8 cm.

### Answer: 8 cm

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 Opus:

Let's approach this step by step:

1) We know that the volume of a cube is given by the formula:

   Volume = edge length^3

2) In this case, the volume is 8 cubic feet. So we can write:

   8 = edge length^3

3) To find the edge length, we need to find the cube root of both sides:

   ∛8 = ∛(edge length^3)

4) The cube root and the power of 3 cancel out on the right side:

   ∛8 = edge length

5) The cube root of 8 is 2, because 2^3 = 8.

   2 = edge length

Therefore, the length of the edge of the cubic hole is 2 feet.

### Answer: 2 feet.

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 Opus:

Let's approach this step by step:

1) Let's define our variables. Let $t$ be the thickness of the string in hundreds of a millimeter. Let $p$ be the pitch in Hz.

2) We are told that the thickness cubed determines the pitch. This means:

   $p = t^3$

3) We are also told that the desired pitch is 343 Hz. So:

   $343 = t^3$

4) To find $t$, we need to take the cube root of both sides:

   $\sqrt[3]{343} = \sqrt[3]{t^3}$

   $7 = t$

5) Remember, $t$ is in hundreds of a millimeter. So the actual thickness is 7 hundreds of a millimeter, or 0.07 millimeters.

### Answer: The thickness of the string should be 0.07 millimeters.

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 Opus:

Let's solve this step by step:

1) We know that the compass box is a cube. In a cube, all edges have the same length. Let's call this length $x$.

2) The volume of a cube is given by the cube of its edge length. So, if the edge length is $x$, the volume is $x^3$.

3) We are told that the volume of the compass box is 729 cubic cm. So we can write an equation:

   $x^3 = 729$

4) To solve for $x$, we need to find the cube root of both sides:

   $\sqrt[3]{x^3} = \sqrt[3]{729}$

   $x = 9$

Therefore, the length of one edge of the cubical compass box is 9 cm.

### Answer: 9 cm

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 Opus:

Let's approach this step by step:

1) We know that the volume of a cube is given by the formula:

   Volume = side length³

2) In this case, the volume of each page (which is in the shape of a cube) is 64 cubic inches. So we can write:

   64 = side length³

3) To find the side length, we need to find the cube root of 64. The cube root of a number is a value that, when cubed (raised to the power of 3), gives the number.

4) The cube root of 64 is 4, because 4³ = 4 × 4 × 4 = 64.

Therefore, the side length of each page is 4 inches.

### Answer: 4 inches.

Expected answer: 4

Correct? Yes