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:

Now, this could look like a difficult problem; however, think of the equation for finding the length of the diagonal of a cube. It is like the Pythagorean Theorem, just adding an additional dimension:

(It is very easy, because the three lengths are all the same: ).

So, we know this, then:

To solve, you can factor out an  from the root on the right side of the equation:

Just by looking at this, you can tell that the answer is:

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 is:

Now, this could look like a difficult problem.  However, think of the equation for finding the length of the diagonal of a cube. It is like the Pythagorean Theorem, just adding an additional dimension:

(It is very easy, because the three lengths are all the same: ).

So, we know this, then:

To solve, you can factor out an  from the root on the right side of the equation:

Just by looking at this, you can tell that the answer is:

Now, use the equation for the volume of a cube:

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

For our data, it is: