Air is being pumped into a spherical balloon at a rate of 500 cubic centimeters per minute.

Question

(Note the volume of a sphere is 43πr3) Determine the rate at which the radius of the balloon is increasing when the radius of the balloon is 20 centimeters.

Answer 1

V(r) = (4/3)pi(r^3) take the derivative of both sides

V'(r) = 4p(r^2)r' V=500, r=20

500=4pi(20)^2(r') = 1600pi(r')

r' = 500/1600pi = 5/16pi cm per minute

change in radius = about 0.1 cm/min

when the radius =20 cm, V= (4/3)pi(20)^3 = (4pi/3)(8000) cm^3 = 32,000pi/3 cm^3

0,.1 or 1/10 of a minute = 6 seconds

in 6 seconds, the Volume will be (4/3)pi(20.1)^3 = 32,482pi/3 cm^3

the volume increased by 482pi/3 = about 500 when the radius increased by 0.1 cm

rate of change of the radius is 0.1 cm per minute when the volume is changing by 500 cm^3 per minute