Question

The radius of a sphere is increasing at a rate of 4 mm/s. How fast is the volume increasing (in mm^3/s) when the diameter is 60 mm

Answer 1

Answer:

The volume is increasing at a rate of 1508 cubic millimeters per second when the diameter is 60 mm.

Step-by-step explanation:

Volume of a sphere:

The volume of a sphere is given by the following equation:

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

In which r is the radius.

Implicit derivatives:

This question is solving by implicit derivatives. We derivate V and r, implicitly as function of t. So

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

The radius of a sphere is increasing at a rate of 4 mm/s.

This means that $\frac{dr}{dt} = 4$

How fast is the volume increasing (in mm^3/s) when the diameter is 60 mm?

This is $\frac{dV}{dt}$ when $r = \frac{d}{2} = \frac{60}{2} = 30$. So

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

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

$\frac{dV}{dt} = 1508$

The volume is increasing at a rate of 1508 cubic millimeters per second when the diameter is 60 mm.