Question

Quick calculus question work doesn't have to be shown

Suppose the radius of a spherical balloon is growing at a rate of 50 cm/s . How quickly is the volume when the radius is 6cm?

Answer 1

`\bf \textit{volume of a sphere}\\\\ V=\cfrac{4\pi r^3}{3}\implies V=\cfrac{4\pi }{3}r^3 \\\\\\ \cfrac{dV}{dt}=\cfrac{4\pi }{3}\cdot \stackrel{chain~rule}{3r^2\cdot \cfrac{dr}{dt}}\quad \begin{cases} (dr)/(dt)=50\\ r=6 \end{cases}\implies \cfrac{dV}{dt}=\cfrac{4\pi }{3}\cdot 3(6)^2\cdot 50\\\\\\ \cfrac{dV}{dt}=\stackrel{(cm^3)/(s)}{7200\pi}`