Difference between revisions of "1950 AHSME Problems/Problem 21"

m (small latex fix)
 
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== Problem==
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== Problem ==
  
 
The volume of a rectangular solid each of whose side, front, and bottom faces are <math> 12\text{ in}^{2} </math>, <math> 8\text{ in}^{2} </math>, and <math> 6\text{ in}^{2} </math> respectively is:
 
The volume of a rectangular solid each of whose side, front, and bottom faces are <math> 12\text{ in}^{2} </math>, <math> 8\text{ in}^{2} </math>, and <math> 6\text{ in}^{2} </math> respectively is:
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<math> \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} </math>
 
<math> \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} </math>
  
==Solution==
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== Solution ==
  
 
If the sidelengths of the cubes are expressed as <math>a, b,</math> and <math>c,</math> then we can write three equations:
 
If the sidelengths of the cubes are expressed as <math>a, b,</math> and <math>c,</math> then we can write three equations:
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<cmath>\begin{align*}
 
<cmath>\begin{align*}
(ab)(bc)(ac)&=(12)(8)(6)\
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(ab)(bc)(ac) &= (12)(8)(6)\
a^2b^2c^2&=576\
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a^2b^2c^2 &= 576\
abc&=\boxed{\mathrm{(B)}\ 24.}
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abc &= \boxed{\mathrm{(B)}\ 24}
 
\end{align*}</cmath>
 
\end{align*}</cmath>
  
==See Also==
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== See Also ==
  
 
{{AHSME 50p box|year=1950|num-b=20|num-a=22}}
 
{{AHSME 50p box|year=1950|num-b=20|num-a=22}}

Latest revision as of 23:54, 11 October 2020

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*}

See Also

1950 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 20
Followed by
Problem 22
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All AHSME Problems and Solutions

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