Bunuel wrote:
Tough and Tricky questions: Geometry.
A rectangular solid box is x inches long, y inches wide and z inches tall, where x, y, and z are positive integers, exactly two of which are equal. What is the total surface area of the box?
(1) One face of the box has an area of 9 square inches.
(2) One face of the box has an area of 81 square inches.
Kudos for a correct solution. Official Solution:A rectangular solid box is x inches long, y inches wide and z inches tall, where x, y, and z are positive integers, exactly two of which are equal. What is the total surface area of the box? To answer the question, we need to know the three dimensions of the box (although we don't need to know which dimension is the length or width or height).
Statement (1): INSUFFICIENT. Since the dimensions of the box are integers, the possible dimensions of 2 of the sides are either (1, 9) or (3, 3). In the first case, the third dimension of the box must be either 1 or 9 (to make two of the dimensions the same). In the second case, the third dimension must be any positive integer other than 3 (to prevent all three dimensions from being equal). We do not know enough to get the third dimension, however.
Statement (2): INSUFFICIENT. Since the dimensions of the box are integers, the dimensions of 2 of the sides could be (1, 81), (3, 27), or (9, 9). In the first case, the third dimension of the box must be either 1 or 81; in the second case, the third dimension must be 3 or 27. In the third case, the third dimension must be any positive integer other than 9. Again, we do not know enough to get the third dimension.
Statements (1) & (2): INSUFFICIENT. Using the foregoing, we can construct two cases that satisfy all the criteria: (1, 9, 9) and (3, 3, 27). These two cases lead to different surface areas (\(9+9+9+9+81+81=198\) sq. inches and \(81+81+81+81+9+9=342\) sq. inches).
Answer: E.
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