Difference between revisions of "2015 AMC 10A Problems/Problem 9"

(Created page with "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 th...")
 
(Added answer choices, See Also, and Solution.)
Line 1: Line 1:
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?
+
==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 $10\%$ 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

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