The volume of a sphere is given by
`V= 4/3 pi R^3`
where R is the radius of the sphere (a). Find the rate of change of volume with respect to R. (b). Find the change in volume of the sphere as the radius is increased from 20.0 cm to 20.1 cm. Assume that the rate does not appreciably change between R=20.0 cm to R=20.1 cm.

Explanation

a. `V= 4/3 pi R^3`
or, `(dV)/(dR) = 4/3 pi d/dr (R)^3 = 4/3 pi . 3 R^3=4 pi R^2`.
b. At R=20 cm, the rate of change of volume with the radius is
`(dV)/(dR)= 4 pi R^2= (4 pi (400 cm^2)`
=1600 pi cm^2`
`= 160 pi cm^3`