2014-07-21 09:00:40

In a certain rectangular prism, at least two faces have an area of 24, and at least two faces have an area of 16. If all the sides of the prism have lengths that are integer values, which of the following could be the total volume of the prism?

I. 48 cubic units
II. 96 cubic units
III. 192 cubic units

(A) I only
(B) I and II
(C) I and III
(D) II and III
(E) I, II, and III

The correct answer is E
Explanation: Start by figuring out integer lengths for the given sides by looking for factors of them.

The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
The factors of 16 are 1, 2, 4, 8, and 16.

Next, check for possible ways of making cubes with these numbers by looking for similar factors in the answer choices (basically three of the factors need to be similar).

48 has factors of 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. This means the prism could have sides of 2, 3, and 8. That would result in two faces of 6, two faces of 16, and and two faces of 24. It would also result in a volume of 2 x 3 x 8, or 48.

96 has factors of 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, and 96. This means the prism could have sides of 4, 4, and 6. If the sides are 4, 4, and 6, the result would be two faces of 16, two of 24, and another set of 24. The volume of that prism would be 4 x 4 x 6, or 96 cubic units.

192 has factors of 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, and 192. This means the prism could have sides of 2, 8, and 12. If it did, the prism would have two sides of area 16, two sides of area 24, and two sides of area 96. It would also have a volume of 2 x 8 x 12, or 192.

Since I, II, and III are all possible, choice E is correct.