Difference between revisions of "2015 AMC 10A Problems/Problem 9"
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| − | Two right circular cylinders have the same volume. | + | ==Problem== |
| + | Two right circular cylinders have the same volume. The radius of the second cylinder is <math>10\%</math> more than the radius of the first. What is the relationship between the heights of the two cylinders? | ||
| + | |||
| + | <math>\textbf{(A)}\ \text{The second height is } 10\% \text{ less than the first.} \\ \textbf{(B)}\ \text{The first height is } 10\% \text{ more than the second.}\\ \textbf{(C)}\ \text{The second height is } 21\% \text{ less than the first.} \\ \textbf{(D)}}\ \text{The first height is } 21\% \text{ more than the second.}\\ \textbf{(E)}\ \text{The second height is } 80\% \text{ of the first.}</math> | ||
| + | |||
| + | ==Solution== | ||
| + | Let the radius of the first cylinder be <math>r_1</math> and the radius of the second cylinder be <math>r_2</math>. Also, let the height of the first cylinder be <math>h_1</math> and the height of the second cylinder be <math>h_2</math>. We are told <cmath>r_2=\frac{11r_1}{10}</cmath> <cmath>\pi r_1^2h_1=\pi r_2^2h_2</cmath> Substituting the first equation into the second and dividing both sides by <math>\pi</math>, we get <cmath>r_1^2h_1=\frac{121r_1^2}{100}h_2\implies h_1=\frac{121h_2}{100}.</cmath> Therefore, <math>\boxed{\textbf{(D)}\ \text{The first height is } 21\% \text{ more than the second.}}</math> | ||
| + | |||
| + | ==See Also== | ||
| + | {{AMC10 box|year=2015|ab=A|num-b=8|num-a=10}} | ||
| + | {{MAA Notice}} | ||
Revision as of 17:03, 4 February 2015
Problem
Two right circular cylinders have the same volume. The radius of the second cylinder is 
more than the radius of the first. What is the relationship between the heights of the two cylinders?
$\textbf{(A)}\ \text{The second height is } 10\% \text{ less than the first.} \\ \textbf{(B)}\ \text{The first height is } 10\% \text{ more than the second.}\\ \textbf{(C)}\ \text{The second height is } 21\% \text{ less than the first.} \\ \textbf{(D)}}\ \text{The first height is } 21\% \text{ more than the second.}\\ \textbf{(E)}\ \text{The second height is } 80\% \text{ of the first.}$ (Error compiling LaTeX. Unknown error_msg)
Solution
Let the radius of the first cylinder be 
and the radius of the second cylinder be 
. Also, let the height of the first cylinder be 
and the height of the second cylinder be 
. We are told ![\[r_2=\frac{11r_1}{10}\]](http://latex.artofproblemsolving.com/4/1/7/41704dff30bf2daea06efcba03fdcd72cb507478.png)
![\[\pi r_1^2h_1=\pi r_2^2h_2\]](http://latex.artofproblemsolving.com/1/7/7/177bac559181a4d8c72c9057a70337eabab64b2f.png)
Substituting the first equation into the second and dividing both sides by 
, we get ![\[r_1^2h_1=\frac{121r_1^2}{100}h_2\implies h_1=\frac{121h_2}{100}.\]](http://latex.artofproblemsolving.com/c/e/3/ce3ff9714fedab21b6406945de393a33855a715f.png)
Therefore, 
See Also
| 2015 AMC 10A (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 10 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
