Find rate that the volume changes

Question

The volume V=4/3πr³

of a spherical balloon changes with the radius.

a. At what rate (ft³/ft​) does the volume change with respect to the radius when r=1 ft?

b. Using the rate from part a​, by approximately how much does the volume increase when the radius changes from 1 to 1.5 ft?

Answer 1

V'(r) = 4pir^2

V'(1) = 4pi

V'(1.5) =4pi(2.25) = 9pi

V(1) = (4/3)pi

V(1.5)= (4/3)pi(1.5)^3 = 4.5pi

V(1.5)-V(1) -=(4.5-4/3)pi = (19/6)pi ft^3 increase from r=1 to 1.5

a little over 3 cubic feet increase with the half foot increase in radius