# word problem

• Jun 10th 2009, 08:24 PM
jimmyp
word problem
The radius $\displaystyle r$ of a right circular cone is increasing at a rate of 2 inches per minute. The height $\displaystyle h$ of the cone is related to the radius by $\displaystyle h=3r$. Find the rates of change of the volume when :

$\displaystyle r$ = 6 inches
and
$\displaystyle r$ = 24 inches

How can I set up the derivative and solve down? (step by step instruction works best for me :))
• Jun 10th 2009, 08:31 PM
TheEmptySet
Quote:

Originally Posted by jimmyp
The radius $\displaystyle r$ of a right circular cone is increasing at a rate of 2 inches per minute. The height $\displaystyle h$ of the cone is related to the radius by $\displaystyle h=3r$. Find the rates of change of the volume when :

$\displaystyle r$ = 6 inches
and
$\displaystyle r$ = 24 inches

How can I set up the derivative and solve down? (step by step instruction works best for me :))

$\displaystyle V=\frac{1}{3}\pi r^2 h$

Taking the derviative with respect to time gives

$\displaystyle \frac{dV}{dt}=\frac{\pi}{3}\left(2rh \frac{dr}{dt}+r^2\frac{dh}{dt} \right)$

From above we know that $\displaystyle h=3r \implies \frac{dh}{dt}=3\frac{dr}{dt}$

From here just plug everything in.

Good luck