Question

Air is being pumped into a spherical balloon at a rate of 5 cm^3/min. Determine the rate at which the radius of the balloon is increasing when the diameter of the balloon is 20 cm

Answer 1

0.08 cm/min

Step-by-step explanation:

Given:

$\dfrac{dV}{dt}=5\:\text{cm}^3\text{/min}$

Find $\frac{dr}{dt}$ when diameter D = 20 cm.

We know that the volume of a sphere is given by

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

Taking the time derivative of V, we get

$\dfrac{dV}{dt} = 4\pi r^2\dfrac{dr}{dt} = 4\pi\left(\dfrac{D}{2}\right)^2\dfrac{dr}{dt} = \pi D^2\dfrac{dr}{dt}$

Solving for $\frac{dr}{dt}$, we get

$\dfrac{dr}{dt} = \left(\dfrac{1}{\pi D^2}\right)\dfrac{dV}{dt} = \dfrac{1}{\pi(20\:\text{cm}^2)}(5\:\text{cm}^3\text{/min})$

$\:\:\:\:\:\:\:= 0.08\:\text{cm/min}$