Since the front face of the prism is a square, the common sidelength - and the missing dimension - is .

There are two faces (front and back) that are squares of sidelength ; the area of each is the square of this, or .

There are four faces (left, right, top, bottom) that are rectangles of dimensions 25 and ; the area of each is the product of the two, .

The surface area is the total of their areas:

Since the top face of the prism is a square, the common sidelength - and the missing dimension - is 25.

The surface area  of a rectangular prism with length , width , and height  can be found using the formula

.

Setting, and solving for :

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 question tells us that the cube is split into two prisms of equal size. Therefore, we don't even need to use the formula for the volume of a prism. If the cube is split into two equal forms, then the two prisms will be equal to one-half of the cube's volume.

We know that the cube has a volume of 27 cubic inches, so each side of the cube must be ∛27=3 inches. Since the cube is inscribed inside the sphere, the diameter of the sphere is the diagonal length of the cube, so the radius of the sphere is half of the diagonal length of the cube. To find the diagonal length of the cube, we use the distance formula d=√(3²+3²+3² )=√(3*3² )=3√3, and then divide the result by 2 to find the radius of the sphere, (3√3)/2.

S = 4π(r²)

100π = 4π(r²)

100 = 4r²

25 = r²

5 = r

In order to find the radius, first write the surface area formula for a sphere and substitute the surface area.

Divide by  on both sides in order to isolate the  term.

Simplify both sides of the equation.

Square root both sides.

The radius is .