Based on our prompt, we can say that the prism has dimensions that can be represented as:

Dim1: x

Dim2: 2 * Dim1 = 2x

Dim3: (1/4) * Dim2 = (1/4) * 2x = (1/2) * x

More directly stated, therefore, our dimensions are: x, 2x, and 0.5x. Therefore, the volume is x * 2x * 0.5x = 64, which simplifies to x³ = 64. Solving for x, we find x = 4. Therefore, our dimensions are:

x = 4

2x = 8

0.5x = 2

Or: 2 x 4 x 8

Based on our prompt, we can say that the prism has dimensions that can be represented as:

Dim1: x

Dim2: 3 * Dim1 = 3x

Dim3: 5 * Dim1 = 5x

More directly stated, therefore, our dimensions are: x, 3x, and 5x. Therefore, the volume is x * 3x * 5x = 120, which simplifies to 15x³ = 120 or x³ = 8. Solving for x, we find x = 2. Therefore, our dimensions are:

x = 2

3x = 6

5x = 10

Or: 2 x 6 x 10

In order to figure out how many cubic centimeters can fit into the box, we need to figure out the volume of the box in terms of cubic centimeters. However, the measurements of the box are given in meters. Therefore, we need to convert these measurements to centimeters and then determine the volume of the box.

There are 100 centimeters in one meter. This means that in order to convert from meters to centimeters, we must multiply by 100.

The length of the box is 2 meters, which is equal to 2 x 100, or 200, centimeters.

The width of the box is 0.5(100) = 50 centimeters.

The height of the box is 3.2(100) = 320 centimeters.

Now that all of our measurements are in centimeters, we can calculate the volume of the box in cubic centimeters. Remember that the volume of a rectangular box (or prism) is equal to the product of the length, width, and height.

V = length x width x height

V = (200 cm)(50 cm)(320 cm) = 3,200,000 cm³

To rewrite this in scientific notation, we must move the decimal six places to the left.

V = 3.2 x 10⁶ cm³

The answer is 3.2 x 10⁶.

Let l be the length, w be the width, and h be the height of the prism. We are told that the length is twice the width, and that the width is twice the height. We can set up the following two equations:

l = 2w

w = 2h

Next, we are told that the surface area is equal to 252 square units. Using the formula for the surface area of the rectangular prism, we can write the following equation:

surface area = 2lw + 2lh + 2wh = 252

We now have three equations and three unknowns. In order to solve for one of the variables, let's try to write w and l in terms of h. We know that w = 2h. Because l = 2w, we can write l as follows:

l = 2w = 2(2h) = 4h

Now, let's substitute w = 2h and l = 4h into the equation we wrote for surface area.

2(4h)(2h) + 2(4h)(h) + 2(2h)(h) = 252

Simplify each term.

16h² + 8h² + 4h² = 252

Combine h² terms.

28h² = 252

Divide both sides by 28.

h² = 9

Take the square root of both sides.

h = 3.

This means that h = 3. Because w = 2h, the width must be 6. And because l = 2w, the length must be 12.

Because we now know the length, width, and height, we can find the volume of the prism, which is what the question ultimately requires us to find.

volume of a prism = l • w • h

volume = 12(6)(3)

= 216 cubic units

The answer is 216.

The volume of B is 7920 in³. The volume of A is 7722 in³. Treasure Chest B can hold more treasure.

The problem is simple, but be careful. The units are not equal. First convert the last dimension into inches. There are 12 inches per foot. Therefore, the prism's dimensions really are: 4 in x 12 in x 24 in.

From this point, things are relatively easy. The distance from corner to corner in a three-dimensional prism like this can be found by using a variation on the Pythagorean Theorem that merely adds one dimension. That is, d²  = x² + y² + z², or d = √(x² + y² + z²)

For our data, this would be:

d = √(4² + 12² + 24²) = √(16 + 144 + 576) = √(736) = √(2 * 2 * 2 * 2 * 2 * 23) = 4√(46)

First, let's represent our dimensions. We know the bottom could be represented as being x by x. The height is said to be three times one of these dimensions, so let's call it 3x. Based on this, we know the dimensions of the prism are x, x, and 3x. Now, the volume of a right rectangular prism is found by multiplying together its three dimensions. Therefore, if we know the overall volume is 375 in³, we can say:

375 = x * x * 3x or 375 = 3x³

Simplifying, we first divide by 3: 125 = x³. Taking the cube root of both sides, we find that x = 5.

Now, be careful. The dimensions are not 5, 5, 5. They are (recall) x, x, and 3x. If x = 5, this means the dimensions are 5, 5, and 15.

At this point, things are beginning to progress to the end of the problem. The distance from corner to corner in a three-dimensional prism like this can be found by using a variation on the Pythagorean Theorem that merely adds one dimension. That is, d²  = x² + y² + z², or d = √(x² + y² + z²)

For our data, this would be: d = √(5² + 5² + 15²) = √(25 + 25 + 225) = √(275) = √(5 * 5 * 11) = 5√(11) in

First, let's represent our dimensions. We know the bottom could be represented as being x by 3x. The height is said to be twice the longer dimension, so let's call it 2 * 3x, or 6x. Based on this, we know the dimensions of the prism are x, 2x, and 6x. Now, the volume of a right rectangular prism is found by multiplying together its three dimensions. Therefore, if we know the overall volume is 13,122 in³, we can say:

13,122 = x * 3x * 6x or 13,122 = 18x³

Simplifying, we first divide by 18: 729 = x³. Taking the cube root of both sides, we find that x = 9.

Now, be careful. The dimensions are not 9, 9, and 9. They are (recall) x, 3x, and 6x. If x = 9, this means the dimensions are 9, 27, and 54.

At this point, things are beginning to progress to the end of the problem. The distance from corner to corner in a three-dimensional prism like this can be found by using a variation on the Pythagorean Theorem that merely adds one dimension. That is, d²  = x² + y² + z², or d = √(x² + y² + z²)

For our data, this would be: d = √(9² + 27² + 54²) = √(81 + 729 + 2916) = √(3726) = √(2 * 3 * 3 * 3 * 3 * 23) = 9√(46) in.

The diagonal from the top back left corner to the bottom front right corner will be the hypotenuse of a right triangle. The sides of the triangle will be the height of the box and the diagonal through the middle of one of the rectangular faces. We will be able to solve for the length using the Pythagorean Theorem.

To calculate the length of the hypotenuse, we first must find the length of the rectangular diagonal using the sides of the rectangle. This diagonal will be the hypotenuse of a right triangle with sides 7 and 4. Solve for the diagonal length using the Pythagorean Theorem.

Now we can return to our first triangle. We are given the height, 4, and now have the length of the rectangular diagonal. Use these values to solve for the length of the diagonal that connects the top back left corner and the bottom front right corner.