Question

Little Susie is enjoying a nice spherical lollipop. She sucks the lollipop in such away that the circumference decreases by 1 centimeter per minute. How fast is the volume of herlollipop changing when the lollipop has a radius of 5 centimeters? Remember to include units

Answer 1

Answer:

-50 cm^3/min

Explanation:

The volume of the lollipop is given by

$V=\frac{4}{3}\pi r^3$

where r is the radius in centimeters.

To find how fast the volume changes, we need to find the rate of change of the volume, which is given by its derivative with respect to time:

$\frac{dV}{dt}=4\pi r^2 \frac{dr}{dt}$ (1)

where

dr/dt is the rate of change of the radius

We know that the circumference is

$C=2\pi r$

and it decreases by 1 cm per minute, so

$\frac{dC}{dt}=2\pi \frac{dr}{dt}=-1$ (cm/min)

from which we find dr/dt:

$\frac{dr}{dt}=-\frac{1}{2\pi}$

Substituting into (1),

$\frac{dV}{dt}=4\pi r^2 \frac{1}{2\pi}=-2r^2$  (cm^3/min)

And substituting r = 5, we find the rate of change of the volume when the radius is 5 cm:

$\frac{dV}{dt}=-2(5)^2=-50$ (cm^3/min)