Question

What is the volume of a certain rectangular solid?
(1) Two adjacent faces of the solid have areas 15 and 24, respectively.
(2) Each of two opposite faces of the solid has area 40.

• A. Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
• C. Both statements together are sufficient, but neither statement alone is sufficient.
• D. Each statement alone is sufficient.
• E. Statements (1) and (2) together are not sufficient.

Answer 1

The correct option is C Both statements together are sufficient, but neither statement alone is sufficient.

1. If the edge lengths of the rectangular solid are 3, 5, and 8, then two adjacent faces will have areas 3 * 5 = 15 and 3 * 8 = 24 and the volume of the rectangular solid will be 3 * 5 * 8 = 120. If the edge lengths of the rectangular solid are 1, 15, 24, then two adjacent faces will have areas 1 * 15 = 15 and 1 * 24 = 24 and the volume of the rectangular solid will be 1 * 15 * 24 = 360; NOT sufficient.
2. If the edge lengths of the rectangular solid are 5, 8, and x, where x is a positive real number, then the rectangular solid will have a pair of opposite faces of area 40, namely the two faces that are 5 by 8. However, the volume is 5 * 8 * x, which will vary as x varies; NOT sufficient.

• Taking (1) and (2) together, if the edge lengths are denoted by x, y, and z, then xy = 15, xz = 24, and yz = 40 and so xy * yz * xz = 15 * 24 * 40, or . Thus, the volume of the rectangular solid is . Therefore, (1) and (2) together are sufficient.
• The correct answer is C; both statements together are sufficient