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:

$2$

$4$

$6$

$6\pi$

$8$

Answer: C

Concepts:cylinder volume quadratic

Difficulty rating: 1770

Small Hint:

Let the original radius be $r$ and write each of the two volume increases

Big Hint:

Equate $2\pi((r+6)^2-r^2)$ and $6\pi r^2$

Solution:

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.

1952 AMC 12 Problem 41

Answer: C

Difficulty rating: 1770

Small Hint:

Big Hint:

Solution:

Problem 41 in Other Years