Question

The radius of a sphere is increasing at a rate of 5 mm/s. How fast is the volume increasing when the diameter is 40mm? Note: The volume of a sphere of radius, r, is given by V = (4/3)pi r^3

Answer 1

Answer:

The volume is increasing at a rate of 25,133 mm/s.

Step-by-step explanation:

Diameter is 40mm

Radius is half the diameter, so $r = \frac{40}{2} = 20$

How fast is the volume increasing when the diameter is 40mm?

We have to apply implicit differentiation, of V and r in function of t. So

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

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

The radius of a sphere is increasing at a rate of 5 mm/s.

This means that $\frac{dr}{dt} = 5$

Then

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

$\frac{dV}{dt} = 4\pi (20)^2(5)$

$\frac{dV}{dt} = 25133$

The volume is increasing at a rate of 25,133 mm/s.