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

Answer:

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

Step-by-step explanation:

Volume of a sphere:

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

$V = \frac{4\pi r^3}{3}$

In this question:

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

$\frac{dV}{dt} = 4\pi r^2\frac{dr}{dt}$

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

This means that $\frac{dr}{dt} = 3$

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

Radius is half the diameter, so $r = 30$. We have to find $\frac{dV}{dt}$. So

$\frac{dV}{dt} = 4\pi r^2\frac{dr}{dt}$

$\frac{dV}{dt} = 4\pi (30)^2(3) = 33929.3$

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