1952 AMC 12 Problem 41

Question

Increasing the radius of a cylinder by

6

units increases the volume by

y

cubic units. Increasing the altitude of the cylinder by

6

units also increases the volume by

y

cubic units. If the original altitude is

2,

then the original radius is:

2

4

6

6\pi

8

Let the original radius be

r

and write each of the two volume increases

Equate

2\pi((r+6)^2-r^2)

and

6\pi r^2

With original height

2

and radius

r,

increasing the radius adds

2\pi\bigl((r+6)^2-r^2\bigr)=24\pi r+72\pi

cubic units. Increasing the height by

6

adds

6\pi r^2

cubic units. Equating these gives

6r^2=24r+72,

or

r^2-4r-12=0.

Its positive root is

r=6.

Thus, the correct answer is C.

Problem 41 in other years