What is the surface area of a certain rectangular solid? (1) Two adja

Official Explanation

To find the surface area of this rectangular solid, we will most likely require the height, width, and depth of the shape. We can call the three dimensions, a, b, and c, so that the surface area would be 2ab + 2ac + 2bc. On to the statements, separately first.

Statement (1) tells us about two adjacent faces. They have one dimension in common, which means they have one variable in common, which we can take to be a. We can interpret this as ab = 30 and ac = 45. That gives us a total of two equations and three variables, so we cannot solve directly. Meanwhile, we don't know bc. We can imagine a case in which a = 5, b = 6, c = 9. Then bc = 54. Case II: a = 15, b = 2, c = 3. Then bc = 6, and since the two cases have different bc's and equal other areas, they have different surface areas. Insufficient.

Statement (2) tells us that abc = 270. That's 30*9, so we could have a = 5, b = 6, c = 9. In another case, we could have a = 27, b = 10, c = 1. In one case, half the surface area is ab + ac + bc = 30 + 45 + 54 = 129. In the other, half the surface area is ab + ac + bc = 270 + ... and we will just stop there because it is already greater, so we have two possibilities. Statement (2) alone is insufficient.

When we combine the statements, we now have three equations and three variables. We will be able to solve completely, and the single possible case turns out to be Case I above, which is not a complete surprise as it was allowed by both statements.

The correct answer is (C).