Question

The radius of a sphere is increasing at a rate of 5 mm/s. How fast is the volume increasing when the diameter is 40mm? Note: The volume of a sphere of radius, r, is given by V = (4/3)pi r^3

Answer 1

The volume is increasing at a rate of 25,133 mm/s.

Explanation:

Diameter is 40mm

Radius is half the diameter, so
`r = (40)/(2) = 20`

How fast is the volume increasing when the diameter is 40mm?

We have to apply implicit differentiation, of V and r in function of t. So

`V = (4)/(3) \pi r^3`

`(dV)/(dt) = 4\pi r^2 (dr)/(dt)`

The radius of a sphere is increasing at a rate of 5 mm/s.

This means that
`(dr)/(dt) = 5`

Then

`(dV)/(dt) = 4\pi r^2 (dr)/(dt)`

`(dV)/(dt) = 4\pi (20)^2(5)`

`(dV)/(dt) = 25133`

The volume is increasing at a rate of 25,133 mm/s.