Question

The radius r of a spherical balloon is expanding at the constant rate of 14 in/min. (a) Determine the rate at which the volume V changes with respect to time, in in3 /min, when r = 8 in. Round your answer to the nearest integer. Recall V = 4 3 πr3 .

Answer 1

Volume of spherical balloon of radius r is given by :

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

Rate of change of radius, $\dfrac{dr}{dt}=14 \ in/min$ .

Now, rate of change of volume is given by :

$\dfrac{dV}{dt}=\dfrac{4}{3}\times \pi (2r^2)\times \dfrac{dr}{dt}\\\\\dfrac{dV}{dt}=\dfrac{8}{3}\pi r^2\times 14\\\\\dfrac{dV}{dt}=\dfrac{112}{3}\pi r^2$

Putting r = 8 in in above equation :

$\left \{ {{y=2} \atop {x=2}} \right. \dfrac{dV}{dt}=\dfrac{112}{3}\times \pi \times 8^2\\\\\dfrac{dV}{dt}=7506.31\ in^2/min$

Hence, this is the required solution.