Printable

2015 AMC 10A Problem 9

2015 AMC 10A Problem 9AMC 10 Step by Step

Name: Date: September 13, 2026

Ratios, Percents & Averages · difficulty 2 of 5 · about 2 min · position P1-10

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?

A)
$\text{The second height is } 10\% \text{ less than the first.}$
B)
$\text{The first height is } 10\% \text{ more than the second.}$
C)
$\text{The second height is } 21\% \text{ less than the first.}$
D)
$\text{The first height is } 21\% \text{ more than the second.}$
E)
$\text{The second height is } 80\% \text{ of the first.}$

Answer

Key insight

Equal volumes mean r^2 h is constant; radius scaled by 1.1 scales r^2 by 1.21, so the first height equals 1.21 times the second.

Solution

Let the first cylinder have radius $r$ and height $h_1$ , and the second have radius $1.1r$ and height $h_2$ . Equal volumes give
$\pi r^2 h_1 = \pi (1.1r)^2 h_2 = 1.21\,\pi r^2 h_2,$
so $h_1 = 1.21\, h_2$ . The first height is $21\%$ more than the second.

Note this is not the same as the second being $21\%$ less than the first: $h_2 = h_1 / 1.21 \approx 0.826\, h_1$ , a decrease of about $17.4\%$ .

The answer is $\boxed{\textbf{(D)}\ \text{The first height is } 21\% \text{ more than the second.}}$

Why this works

With volume fixed, height is inversely proportional to the square of the radius. A $10\%$ increase in radius is a factor $1.1$ , and squaring gives $1.21$ . The remaining care is in the wording: " $A$ is $21\%$ more than $B$ " means $A = 1.21B$ , which is a different statement from " $B$ is $21\%$ less than $A$ ."

Alternative approach

Plug in numbers: first cylinder radius $10$ , second radius $11$ . To match volumes, $100 h_1 = 121 h_2$ ; with $h_2 = 100$ we get $h_1 = 121$ , visibly $21\%$ more than $100$ .

The trap

Answering that the second height is 21% less than the first; 1/1.21 is about 0.826, a 17.4% decrease, so the percent must be stated the other way.

Common mistakes

Answering that the second height is $21\%$ less than the first; $1/1.21 \approx 0.826$ , a $17.4\%$ decrease, so the percent must be stated the other way.
Forgetting to square the radius factor and choosing a $10\%$ relationship, choice (A) or (B).

Techniques

Set up the equation/formula and compute; no special trick needed · Test small/specific values or special cases to find or verify the answer