Question

The radius of a sphere is increasing at a rate of 2 mm/s. How fast is the volume increasing (in mm3/s) when the diameter is 40 mm

Answer 1

`(dV)/(dt)=502.65\ mm^3/s`

Explanation:

The volume of a sphere is given by :

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

The rate of change of volume means,

`(dV)/(dt)=(d)/(dt)((4)/(3)\pi r^3)\\\\(dV)/(dt)=(4)/(3)\pi * 3r* (dr)/(dt)`

We have,
`(dr)/(dt)=2\ mm/s\ and\ r=40\ mm`

So,

`(dV)/(dt)=(4)/(3)\pi * 3* 20* 2\\\\=502.65\ mm^3/s`

So, the volume is increasing at the rate of
`502.65\ mm^3/s`.