A rectangular solid box is x inches long, y inches wide and z inches

Question

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 .

DangerPenguin · Answer

statement 1: insufficient

2 of the values could be 3 and 3 or 9 and 1 but we still don't know what the 3rd length is.

statement 2: insufficient

insufficient for the same reason as above

combined, we still do not know. 1 set of values that works for both statements is 3x3x27 and another is 1x9x9 both which have different surface areas.

Answer E!

akumar5 · Answer

Given that,
length=x;width=y;height=z & exactly two of them is equal

1) Area of one face is 9
 this can be due to 3*3 or 9*1
 So INSUFFICIENT

2)Area of one face is 81
 his can be due to 9*9 or 27*3 or etc..
so INSUFFICIENT and does not give a definite result.

Also no fruitful combination appears.

Answer is E

Bunuel · Answer

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.

TheMechanic · Answer

From Stmt 1: We get sides could be (3,3,1) or (9,1,1) or (3,3,x) where x could by any positive integer value. Not sufficient.
From Stmt 2: We get possible sides as (27,3,3) or (9,9,1) or (81,1,1). Not sufficient.

Combining 1 and 2, we still have (27,3,3) or (9,9,1) satisfying the conditions. They both lead to different surface areas. Hence E.

bumpbot · Answer

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A rectangular solid box is x inches long, y inches wide and z inches

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