**If a rectangular box has two faces, each with an area of 30, two faces with areas of 60 each, and two faces with areas of 72 each, what is its volume?**

**A) 60**

**B) 90**

**C) 162**

**D) 300**

**E) 360**

The best way to solve this is to remember that you could produce three simultaneous equations involving the three variables length (L), height (H) and width (W).

These would be:

L*W = 30

→ L = 30 / W

W* H = 72

→ H = 72 / W

L*H = 60

→ (30 / W) *(72 / W) = 60

→ (30* 72) / 60 = W^2

→ 72 / 2 = W^2

→ 36 = W^2

→ W = 6

Now simply plug in the value of 6 for W to one of the earlier equations to obtain a value for L:

L*W = 30

→ L*6 = 30

→ L = 5

In the same way, plug in the value of 5 for L into one of the earlier equations to obtain a value for H:

L*H = 60

→ 5*H = 60

→ H = 12

The last step to multiply L *W*H to obtain the volume:

5*6*12 = 360

**Therefore, answer choice E is correct.**

**A second strategy for solving this question: Plugging In**

Though we do not know the dimensions of the rectangular box, we can try plugging in potential values for length, width, and height, and see if they give us the required areas of the sides.

We are told that two of the faces each have an area of 30. We can think of the dimensions as possibly being equal to 5 and 6, since 5*6 = 30, or 3 and 10, since 3*10 = 30. (2 and 15 or 1 and 30 could be additional paired possibilities.)

We are also told that another two faces each have an area of 60. We can think of the dimensions as possibly being equal to 5 and 12, or 6 and 10, among others. Of course, one of these edges would also have to work with the previous area of 30. Notice that there are potential overlaps with 5 and 6.

Then there are two faces each with an area of 72. The dimensions creating this could be 8 and 9, but more likely are going to be 6 and 12, since we’ve seen possibilities of 6 and 12 when testing values for the other faces.

Taking overlapping values, we can try 5, 6 and 12 as the dimensions, and do some testing:

5*6 = 30

5*12 = 60

6*12 = 72

All three of these measurements work with the restrictions given in the question, so they must be equal to the length, width, and height of the rectangular box.

We can now calculate the volume: 5*6*12 = 360, which is the correct answer.

Notice you have combined two major areas of math to solve this question. The question appears to be a geometry one, but you can use the rules of algebra or plugging in to help you to solve it.

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