Difference between revisions of "1950 AHSME Problems/Problem 21"
m (small latex fix) |
|||
Line 1: | Line 1: | ||
− | == Problem== | + | == 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: | ||
Line 5: | Line 5: | ||
<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== | + | == 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: | ||
Line 14: | Line 14: | ||
<cmath>\begin{align*} | <cmath>\begin{align*} | ||
− | (ab)(bc)(ac)&=(12)(8)(6)\ | + | (ab)(bc)(ac) &= (12)(8)(6)\ |
− | a^2b^2c^2&=576\ | + | a^2b^2c^2 &= 576\ |
− | abc&=\boxed{\mathrm{(B)}\ 24 | + | abc &= \boxed{\mathrm{(B)}\ 24} |
\end{align*}</cmath> | \end{align*}</cmath> | ||
− | ==See Also== | + | == 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 , , and respectively is:
Solution
If the sidelengths of the cubes are expressed as and then we can write three equations:
The volume is Notice symmetry in the equations. We can find my multiplying all the equations and taking the positive square root.
See Also
1950 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 20 |
Followed by Problem 22 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.