A circular cylinder is inscribed in a fixed sphere with radius 18cm such that all the points on the circ of both cylindrical ends are always lying on the spherical surface. Given that the cylinder has a ht of h cm and cylindrical base radius of r cm, show that $\displaystyle h^2+4r^2=1296$ (done)

Initially the ht of the cylinder is 24cm and is decreasing at constant rate of 0.25cm/s

(ii) show that rate at which radius is changing at this instant is $\displaystyle \frac {\sqrt5} {20} $ cm/s (done)

(iii) Find the time taken for the volume of the cylinder to reach its maximum value. (you need not verify that the volume is maximum) Ans:12.9s

ok part (iii) is the one i need help in. highlight beside it for ans.

i know that max value for volume means $\displaystyle \frac {dv} {dr}=0$ but then i'll just get r=0 when i sub in h=24.

then if i use the h and r eqn, i just get r=0 again.

in truth i'm clueless how to continue... is max volume just taking the given values of h and r, then using that divide by $\displaystyle \frac {dv} {dt} $? if so then i couldn't find that either :/ maybe i'm just thinking too much x.x