1950 AMC 12 Problem 21

Attempt Problem 21 of the 1950 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1950 AMC 12 solutions, or check the answer key.

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21.

The volume of a rectangular solid each of whose side, front, and bottom faces are

12

square inches,

8

square inches, and

6

square inches respectively is:

576

cubic inches

24

cubic inches

9

cubic inches

104

cubic inches

None of these

Let the three edge lengths be

x,

y,

and

z

Multiplying the three face areas gives

(xyz)^2

If the edge lengths are

x,

y,

and

z,

then the three face areas are

xy,

yz,

and

xz.

Therefore

\begin{aligned} (xyz)^2&=(xy)(yz)(xz)\\ &=12\cdot8\cdot6=576. \end{aligned}

Since the volume is positive,

xyz=\sqrt{576}=24

cubic inches.

Thus, the correct answer is B.

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