1951 AMC 12 Problem 23

Attempt Problem 23 of the 1951 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1951 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

23.

The radius of a cylindrical box is

8

inches and the height is

3

inches. The number of inches that may be added to either the radius or the height to give the same nonzero increase in volume is:

1

5\dfrac13

Any number

Non-existent

None of these

Compare the volume after adding

x

to the radius with the volume after adding

x

to the height

Set

\pi(8+x)^2(3)=\pi(8)^2(3+x)

and discard

x=0

For the two new volumes to be equal,

3\pi(8+x)^2=64\pi(3+x).

Cancelling

\pi

and expanding gives

3x^2-16x=0,

x=0

x=\frac{16}{3}=5\dfrac13.

The problem requires a nonzero increase.

Thus, the correct answer is B.

Problem 23 in other years