Answer:
[tex]\mathbf{ \dfrac{dr}{dt} = 0.03730 \ cm/min}[/tex]
Step-by-step explanation:
The rate of the inflation of the balloon with time can be denoted as:
[tex]\dfrac{dv}{dt} = 500 \ cm^3/m[/tex]
To determine; how fast does the radius change with time.
i.e.
[tex]\dfrac{dr}{dt}=???[/tex]
where r = 40 cm and the volume of sphere = [tex]\dfrac{4}{3} \pi r^2[/tex]
∴
[tex]\dfrac{dv}{dt}= \dfrac{4}{3} \pi 2(r^3) \dfrac{dr}{dt}[/tex]
[tex]500= \dfrac{4}{3} \pi \times 2(40^2) \dfrac{dr}{dt}[/tex]
[tex]500= 13404.13 \dfrac{dr}{dt}[/tex]
[tex]\dfrac{500}{13404.13} = \dfrac{dr}{dt}[/tex]
[tex]\mathbf{ \dfrac{dr}{dt} = 0.03730 \ cm/min}[/tex]
