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 100 mm

Answer 1

Answer:

The radius is increasing at a rate of 62832 cubic millimeters per second when the diameter is of 100 mm.

Step-by-step explanation:

Volume of a sphere:

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

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

How fast is the volume increasing:

To find this, we have to differentiate the variables of the problem, which are 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 2 mm/s.

This means that $\frac{dr}{dt} = 2$

How fast is the volume increasing (in mm3/s) when the diameter is 100 mm?

Radius is half the diameter, so $r = \frac{100}{2} = 50$

Then

$\frac{dV}{dt} = 4\pi r^2\frac{dr}{dt} = 4\pi (50)^2(2) = 62832$

The radius is increasing at a rate of 62832 cubic millimeters per second when the diameter is of 100 mm.