## What is the volume of a…

## Solution:

We need to determine the volume of a rectangular solid. If L = length, W = width and H = height, the formula for the volume of a rectangular solid is:

volume of a rectangle solid = length x width x height

**Statement One Alone: **

Two adjacent faces have areas 15 and 24.

Let’s express the information from statement one in a diagram.

Notice that we have labeled the length, width, and height in no particular order. Thus, we know the following:

W x H = 15

L x W = 24

Using these two equations, we can substitute in some convenient numbers to get different sets of values for L, W, and H.

For example, if W = 3, H = 5 and L = 8, then:

Volume = L x W x H = 8 x 3 x 5 = 120

If W = 1, H = 15 and L = 24, then:

Volume = L x W x H = 24 x 1 x 15 = 360

Because we have two different values for the volume, statement one is not sufficient to answer the question. We can eliminate answer choices A and D.

**Statement Two Alone: **

Each of two opposite faces of the solid has area 40.

[Note: Before we begin, statement two is not worded as accurately as it could be. It could mislead readers to think that each of any two opposite faces of the solid has area 40, which would mean, the solid must be a cube, and its volume is determinable when the area of one face is given. However, we think the statement really means “Each of two opposite faces of the solid, and only those two faces, has area 40.” A better statement is: Two opposite faces of the solid each have area 40.]

Let’s display the information from statement two in a diagram.

Since we are given that each of two opposite faces of the solid has area 40, we see that the green face and the red face both have an area of 40. Thus, we can say:

L x H = 40

Because we can get many values for L and H, and because we do not have a value for W, statement two is not sufficient to answer the question. We can eliminate answer choice B.

**Statements One and Two Together: **

Let’s display the information we know from statements one and two in a diagram.

From our two statements we have the following 3 equations:

1) W x H = 15

2) L x W = 24

3) L x H = 40

We see that in selecting some convenient numbers that we can only get one set of values for L, W, and H. That is, L = 8, W = 3, and H = 5.

We can substitute these values into our three equations:

1) W x H = 15

3 x 5 = 15

2) L x W = 24

8 x 3 = 24

3) L x H = 40

8 x 5 = 40

Volume = L x W x H = 8 x 3 x 5 = 120.

Alternatively, we can use algebra to determine the values for W, H, and L. By using algebra, we express all our variables in terms of variable H by manipulating a few of the equations.

1) W x H = 15

2) L x W = 24

3) L x H = 40

Isolating W and L in equations 1 and 3 we have:

W = 15/H

L = 40/H

Since W = 15/H and L = 40/H, we can substitute 40/H for L and 15/H for W in equation 2.

40/H x 15/H = 24

600/H^2 = 24

600 = H^2 x 24

25 = H^2

5 = H

Since H = 5, we see that W = 15/5 = 3 and that L = 40/5 = 8.

Once again, the volume is 8 x 3 x 5 = 120. Both statements together are sufficient.

**Answer:** C