Question

A spherical balloon is inflated with gas at a rate of 500 cubic centimeters per minute. (a) How fast is the radius of the balloon changing at the instant the radius is 40 centimeters

Answer 1

Answer:

$\mathbf{ \dfrac{dr}{dt} = 0.03730 \ cm/min}$

Step-by-step explanation:

The rate of the inflation of the balloon with time can be denoted as:

$\dfrac{dv}{dt} = 500 \ cm^3/m$

To determine; how fast does the radius change with time.

i.e.

$\dfrac{dr}{dt}=???$

where r = 40 cm and the volume of sphere = $\dfrac{4}{3} \pi r^2$

∴

$\dfrac{dv}{dt}= \dfrac{4}{3} \pi 2(r^3) \dfrac{dr}{dt}$

$500= \dfrac{4}{3} \pi \times 2(40^2) \dfrac{dr}{dt}$

$500= 13404.13 \dfrac{dr}{dt}$

$\dfrac{500}{13404.13} = \dfrac{dr}{dt}$

$\mathbf{ \dfrac{dr}{dt} = 0.03730 \ cm/min}$