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 100 mm

Answer 1

Answer:

The Volume of a sphere is given as follows:

V = 4/3*pi*r^3

Implicitly differentiating with respect to time yields:

d/dt(V = 4/3*pi*r^3)

= dV/dt = 4*pi*r^2*dr/dt

You are told in the problem that the radius of the sphere is increasing at a rate of 3mm/sec. That means dr/dt = 3

You also know the diameter is 80mm, meaning the radius is 40mm.

Solve for dV/dt, which is how fast the volume is increasing:

dV/dt = 4*pi*40^2*3

= 19,200*pi mm^3/ sec