## Coffee Can

A coffee company decides to increase the diameter of their cylindrical coffee cans to make the can more appealing to consumers. If the diameter of the can is increased by $25\%$, by what percent will the height be decreased if the volume remains the same?
Source: mathcontest.olemiss.edu 4/8/2013

SOLUTION
Let $r$ be the radius of the original coffee can and $h$ its height.
Volume of original can
$V=\pi r^2h$
Let $r'$ be the radius of the new can and $h'$ its height. If the diameter of the original can is increased by $25\%$, then its radius is also increased by the same amount.
$r'=r+\left (25\%\right )r$
$=r\left (1+.25\right )$
$=1.25r$
Volume of new can
$V'=\pi\left (r'\right )^2h'$
$=\pi\left (1.25r\right )^2h'$
$=1.5625\pi r^2h'$
Volume remains the same
$\pi r^2h=1.5625\pi r^2h'$
Simplify
$h=1.5625h'$
$h'=.64h$
$=(64\%)h$
The new height is $64\%$ of the original height or the original height has decreased by $36\%$.

Answer: $36\%$