Difference between revisions of "2021 Fall AMC 12A Problems/Problem 9"
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~MRENTHUSIASM | ~MRENTHUSIASM | ||
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| + | ==Video Solution by TheBeautyofMath== | ||
| + | https://youtu.be/wlDlByKI7A8?t=649 | ||
| + | |||
| + | ~IceMatrix | ||
==See Also== | ==See Also== | ||
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{{MAA Notice}} | {{MAA Notice}} | ||
Latest revision as of 20:58, 7 April 2022
Problem
A right rectangular prism whose surface area and volume are numerically equal has edge lengths 
and 
What is 
Solution
The surface area of this right rectangular prism is 
The volume of this right rectangular prism is 
Equating the numerical values of the surface area and the volume, we have ![\[2(\log_{2}x\log_{3}x+\log_{2}x\log_{4}x+\log_{3}x\log_{4}x)=\log_{2}x\log_{3}x\log_{4}x.\]](http://latex.artofproblemsolving.com/9/5/1/95157179f7b267173913511a44e13ee5d319d7eb.png)
Dividing both sides by 
we get ![\[2\left(\frac{1}{\log_{4}x}+\frac{1}{\log_{3}x}+\frac{1}{\log_{2}x}\right)=1. \hspace{15mm} (\bigstar)\]](http://latex.artofproblemsolving.com/9/a/2/9a2083b1e94d1b4863154f9646d5246d1526f9fa.png)
Recall that 
and 
so we rewrite 
as
~MRENTHUSIASM
Video Solution by TheBeautyofMath
https://youtu.be/wlDlByKI7A8?t=649
~IceMatrix
See Also
| 2021 Fall AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 8 |
Followed by Problem 10 |
| 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 | |
| All AMC 12 Problems and Solutions | |
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