Since the answer is not dependent on the actual dimensions, for the sake of simplicity, we assume that the base has sidelength 2. Then the area of the base is the square of this, or 4.

The height of the pyramid is equal to the perimeter of the base, or . A right triangle can be formed with the lengths of its legs equal to the height of the pyramid, or 8, and one half the length of a side, or 1; the length of its hypotenuse, which is the slant height, is 

This is the height of one triangular face; its base is a side of the square, so the length of the base is 2. The area of a face is half the product of these dimensions, or 

Since twice the area of the base is , the problem comes down to comparing  and ; the latter, which is (B), is greater.

The volume of a pyramid or a cone with height  and base of area  is 

, 

so in both cases, the area of the base is

Since the pyramid and the cone have the same volume and height, their bases has the same area .

The length of one side of the square base of the pyramid is the square root of this, or , and the perimeter is four times this, or .

The radius and the area of the base of the cone are related as follows:

Multiply both sides by  to get:

, so 

, and

The perimeter of the base of the pyramid, which is (A), is greater than the circumference of the base of the cone.

The cube with the greater sidelength has the greater volume, so we need only calculate and compare sidelengths.

(a) , so the sidelength of the first cube can be found as follows:

 inches

(b)  by an extension of the Pythagorean Theorem, so the sidelength of the second cube can be found as follows:

Since , . The second cube has the greater sidelength and, subsequently, the greater volume. This makes (b) greater.

The sidelengths of Cubes 1, 2, 3, and 4 can be given values , respectively.

Then the volumes of the cubes are as follows:

Cube 1: 

Cube 2: 

Cube 3: 

Cube 4: 

In both answer choices ask for a mean, so we can determine which answer (mean) is greater simply by comparing the sums of volumes.

(a) The sum of the volumes of Cubes 1 and 4 is .

(b) The sum of the volumes of Cubes 2 and 3 is .

Regardless of , the sum of the volumes of Cubes 1 and 4 is greater, and therefore, so is their mean.

This question is relatively straightforward. The equation for the volume of a cube is:

(It is like doing the area of a square, then adding another dimension!)

Now, for our data, we merely need to "plug and chug:"

One of the faces of the cube could be drawn like this:

Notice that this makes a  triangle.

This means that we can create a proportion for the sides. On the standard triangle, the non-hypotenuse sides are both , and the hypotenuse is .  This will allow us to make the proportion:

Multiplying both sides by , you get:

Recall that the formula for the volume of a cube is:

Therefore, we can compute the volume using the side found above:

Now, rationalize the denominator:

The volume of a cube is defined by the formula

where  is the length of one side.

If , then 

and 

So one side measures 7 inches. 

The surface area of a cube is defined by the formula

 , so

The surface area is 294 square inches.

Recall that the formula for the surface area of a cube is:

, where  is the length of a side of the cube. This equation is easy to memorize because it is merely a multiplication of a single side () by  because a cube has  equal sides.

For our data, we know that ; therefore, our equation is:

To solve this, first calculate the side length based on the volume given. Recall that the equation for the volume of a cube is:

, where  is the side length.

For our data, this gives us:

Taking the cube-root of both sides, we get:

Now, use the surface area formula to compute the total surface area:

, where  is the length of a side of the cube. This equation is easy to memorize because it is merely a multiplication of a single side () by  because a cube has  equal sides.

For our data, this gives us:

To solve this, first calculate the side length based on the volume given. Recall that the equation for the volume of a cube is:

, where  is the side length.

For our data, this gives us:

Taking the cube-root of both sides, we get:

(You will need to use a calculator for this.  If your calculator gives you something like  . . . it is okay to round. This is just the nature of taking roots!).

Now, use the surface area formula to compute the total surface area:

, where  is the length of a side of the cube. This equation is easy to memorize because it is merely a multiplication of a single side () by  because a cube has  equal sides.

For our data, this gives us: