Question

The radius of a spherical balloon is increasing at a rate of 2 centimeters per minute. How fast is the volume changing when the radius is 14 centimeters?

Answer 1

`1568*pi` cc/minute

Explanation:

If the radius is r, then the rate of change of r with respect to time t,
`d/dt(r) = 2` cm/minute
Volume as a function of radius r for a spherical object is
`V(r) = 4/3*pi*r^3`
We need to find `d/dt(V)` at r = 14cm

Now, `d/dt(V) = d/dt(4/3*pi*r^3) = (4pi)/3*3*r^2*d/dt(r) = 4pi*r^2*d/dt(r)`

But `d/dt(r)` = 2cm/minute. Thus, `d/dt(V)` at r = 14 cm is:
`4pi*14^2*2` cubic cm / minute `=1568*pi` cc/minute