Question

The radius of a sphere is increasing at a rate of 3 mm/s. How fast is the volume increasing when the diameter is 60 mm

Answer 1

The volume is increasing at a rate of 33929.3 cubic millimeters per second.

Explanation:

Volume of a sphere:

The volume of a sphere of radius r is given by:

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

In this question:

We have to derivate V and r implicitly in function of time, so:

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

The radius of a sphere is increasing at a rate of 3 mm/s.

This means that
`(dr)/(dt) = 3`

How fast is the volume increasing when the diameter is 60 mm?

Radius is half the diameter, so
`r = 30`. We have to find
`(dV)/(dt)`. So

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

`(dV)/(dt) = 4\pi (30)^2(3) = 33929.3`

The volume is increasing at a rate of 33929.3 cubic millimeters per second.