A spherical balloon is being inflated. At time t t\,t seconds the balloon has radius r r\,r cm and volume V V\,V cm3^33, where V=43πr3\displaystyle V = \frac{4}{3}\pi r^3V=34πr3.
The volume of the balloon is modelled as increasing at a constant rate.
Show that
drdt=kr2\displaystyle \frac{dr}{dt} = \frac{k}{r^2}dtdr=r2k
where k k\,k is a positive constant.
The model applies for t>0t>0t>0, with r→0 r\to0\,r→0 as t→0+t\to0^+t→0+. After 5 seconds the radius is 4 cm. Solve the differential equation to find an equation linking r r\,r and ttt.
Suggest one limitation of this model.