Question

Related rates(calculus)

The radius of a sphere is increasing at a rate of 4  mm/s. How fast is the volume increasing when the diameter is 

100  mm?

Answer 1

The volume increases at a rate of
`125663.71 mm^ 3 / s`

Explanation:

The rate of change of the radius with respect to time is 4 mm / s.

So:

`(dr)/(dt) = 4mm / s`

Now we must find a relationship between the volume of a sphere and its radius.

The equation of the volume of a sphere is:

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

So:

`(dV)/(dt) = (4)/(3)\pi * 3r ^ 2 * (dr)/(dt)\\(dV)/(dt) = 4\pi r ^ 2 * 4\\(dV)/(dt) = 16\pi r ^ 2`

The diameter of the sphere is 100 mm. Therefore its radius is 100/2 = 50 mm.

So:

`(dV)/(dt) = 16\pi (50) ^ 2\\(dV)/(dt) = 40000\pi mm^3 / s`

The volume increases at a rate of :

40000 π mm ^ 3 / s = 125663.71 mm^3/s