Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2021 Fall AMC 12A Problems/Problem 9"

(Created page with "==Problem == A right rectangular prism whose surface area and volume are numerically equal has edge lengths <math>\log_{2}x, \log_{3}x,</math> and <math>\log_{4}x.</math> Wha...")
 
Line 1: Line 1:
−
==Problem ==
+
==Problem==
  
 
A right rectangular prism whose surface area and volume are numerically equal has edge lengths <math>\log_{2}x, \log_{3}x,</math> and <math>\log_{4}x.</math> What is <math>x?</math>
 
A right rectangular prism whose surface area and volume are numerically equal has edge lengths <math>\log_{2}x, \log_{3}x,</math> and <math>\log_{4}x.</math> What is <math>x?</math>
−
<math>\textbf{(A)}\ 2\sqrt{6} \qquad\textbf{(B)}\ 6\sqrt{6} \qquad\textbf{(C)}\ 24 \qquad\textbf{(D)} 48 \qquad\textbf{(E)}\ 576</math>
+
 
 +
<math>\textbf{(A)}\ 2\sqrt{6} \qquad\textbf{(B)}\ 6\sqrt{6} \qquad\textbf{(C)}\ 24 \qquad\textbf{(D)}\ 48 \qquad\textbf{(E)}\ 576</math>
  
 
==Solution==
 
==Solution==
−
IN PROGRESS. NO EDIT ON THIS PAGE PLEASE.
+
The surface area of this right rectangular prism is <math>2(\log_{2}x\log_{3}x+\log_{2}x\log_{4}x+\log_{3}x\log_{4}x).</math>
 +
 
 +
The volume of this right rectangular prism is <math>\log_{2}x\log_{3}x\log_{4}x.</math>
  
 +
Equating the numerical values of the surface area and the volume, we have <cmath>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.</cmath>
 +
Dividing both sides by <math>\log_{2}x\log_{3}x\log_{4}x,</math> we get <cmath>2\left(\frac{1}{\log_{4}x}+\frac{1}{\log_{3}x}+\frac{1}{\log_{2}x}\right)=1. \hspace{15mm} (\bigstar)</cmath>
 +
Recall that <math>\log_{b}a=\frac{1}{\log_{a}b}</math> and <math>\log_{b}\left(a^n\right)=n\log_{b}a,</math> so we rewrite <math>(\bigstar)</math> as
 +
<cmath>\begin{align*}
 +
2(\log_{x}4+\log_{x}3+\log_{x}2)&=1 \\
 +
2\log_{x}24&=1 \\
 +
\log_{x}576&=1 \\
 +
x&=\boxed{\textbf{(E)}\ 576}.
 +
\end{align*}</cmath>
 
~MRENTHUSIASM
 
~MRENTHUSIASM
  

Revision as of 19:05, 23 November 2021

Problem

A right rectangular prism whose surface area and volume are numerically equal has edge lengths $\log_{2}x, \log_{3}x,$ and $\log_{4}x.$ What is $x?$

$\textbf{(A)}\ 2\sqrt{6} \qquad\textbf{(B)}\ 6\sqrt{6} \qquad\textbf{(C)}\ 24 \qquad\textbf{(D)}\ 48 \qquad\textbf{(E)}\ 576$

Solution

The surface area of this right rectangular prism is $2(\log_{2}x\log_{3}x+\log_{2}x\log_{4}x+\log_{3}x\log_{4}x).$

The volume of this right rectangular prism is $\log_{2}x\log_{3}x\log_{4}x.$

Equating the numerical values of the surface area and the volume, we have \[2(\log_{2}x\log_{3}x+\log_{2}x\log_{4}x+\log_{3}x\log_{4}x)=\log_{2}x\log_{3}x\log_{4}x.\] Dividing both sides by $\log_{2}x\log_{3}x\log_{4}x,$ we get \[2\left(\frac{1}{\log_{4}x}+\frac{1}{\log_{3}x}+\frac{1}{\log_{2}x}\right)=1. \hspace{15mm} (\bigstar)\] Recall that $\log_{b}a=\frac{1}{\log_{a}b}$ and $\log_{b}\left(a^n\right)=n\log_{b}a,$ so we rewrite $(\bigstar)$ as \begin{align*} 2(\log_{x}4+\log_{x}3+\log_{x}2)&=1 \\ 2\log_{x}24&=1 \\ \log_{x}576&=1 \\ x&=\boxed{\textbf{(E)}\ 576}. \end{align*} ~MRENTHUSIASM

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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2021_Fall_AMC_12A_Problems/Problem_9&oldid=165580"