2015 AMC 10A Problems/Problem 9

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Problem

Two right circular cylinders have the same volume. The radius of the second cylinder is $10\%$ 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 $r_1$ and the radius of the second cylinder be $r_2$. Also, let the height of the first cylinder be $h_1$ and the height of the second cylinder be $h_2$. We are told \[r_2=\frac{11r_1}{10}\] \[\pi r_1^2h_1=\pi r_2^2h_2\] Substituting the first equation into the second and dividing both sides by $\pi$, we get \[r_1^2h_1=\frac{121r_1^2}{100}h_2\implies h_1=\frac{121h_2}{100}.\] Therefore, $\boxed{\textbf{(D)}\ \text{The first height is } 21\% \text{ more than the second.}}$

See Also

2015 AMC 10A (ProblemsAnswer KeyResources)
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

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