Rates and derivatives

Question

While you were out cross country skiing, you make a perfectly shaped spherical snowball. Unfortunately the temperature increases and the volume of your snowball decreases at a rate of -1cc/min. At what rate is the radius decreasing when the radius is 5 cm?

Answer 1

V = (4/3)πr³

 

dV/dt = 4πr²dr/dt

 

dr/dt = (dV/dt) / (4πr²)

 

dV/dt = -1 cc/min

 

r = 5 cm

 

dr/dt = (-1 cc/min) / [4π(5 cm)²] = -[1/(100π)] cm/min

 

The radius decreases at a rate of 1/(100π)] cm/min.