1950 AHSME Problems/Problem 21

Problem

The volume of a rectangular solid each of whose side, front, and bottom faces are $12\text{ in}^{2}$ , $8\text{ in}^{2}$ , and $6\text{ in}^{2}$ respectively is:

$\textbf{(A)}\ 576\text{ in}^{3}\qquad\textbf{(B)}\ 24\text{ in}^{3}\qquad\textbf{(C)}\ 9\text{ in}^{3}\qquad\textbf{(D)}\ 104\text{ in}^{3}\qquad\textbf{(E)}\ \text{None of these}$

Solution

If the sidelengths of the cubes are expressed as $a, b,$ and $c,$ then we can write three equations:

$ab=12, bc=8, ac=6.$

The volume is $abc.$ Notice symmetry in the equations. We can find $abc$ my multiplying all the equations and taking the positive square root.

$\begin{align*} (ab)(bc)(ac)&=(12)(8)(6)\\ a^2b^2c^2&=576\\ abc&=\boxed{\mathrm{(B)}\ 24.} \end{align*}$

1950 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 20	Followed by Problem 22
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1950 AHSME Problems/Problem 21

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