2015 AMC 12A Problem 7

Question

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?

The second height is

10\%

less than the first.

The first height is

10\%

more than the second.

The second height is

21\%

less than the first.

The first height is

21\%

more than the second.

The second height is

80\%

of the first.

Equal volumes give

\pi r^2 h = \pi R^2 H

with

R = 1.1r

Divide by

\pi r^2

to compare

h

and

H;

note

1.1^2 = 1.21

Let

r,h

and

R,H

be the radii and heights of the first and second cylinders. The volumes are equal, so

\pi r^2 h = \pi R^2 H,

and

R = 1.1r.

Then

\pi r^2 h = \pi(1.1r)^2 H

= \pi(1.21 r^2) H.

Dividing by

\pi r^2

yields

h = 1.21H,

so the first height is

21\%

more than the second.

Thus, the correct answer is D.

Problem 7 in other years