Question

Find the height of the right circular cylinder of maximum volume $V$ which can be inscribed in a sphere of radius R.

Answer 1

Radius of Cylinder $=Rcos\theta$
Height of Cylinder $=2Rsin\theta$
Volume of Cylinder $=\pi \left(Rcos\theta {)}^2\right(2Rsin\theta$)



$V=2\pi R^{3}sin\theta co{s}^2\theta$
To maximize volume
$\frac{dV}{d\theta }=0$
$\Rightarrow 2\pi R^3(co{s}^2\theta .cos\theta -2si{n}^2\theta cos\theta )=0$
$co{s}^2\theta .cos\theta -2si{n}^2\theta cos\theta )=0$
$cos\theta (co{s}^2\theta -2si{n}^2\theta )=0$
$cos\theta [co{s}^2\theta -2(1-co{s}^2\theta )=0$
$cos\theta (3co{s}^2\theta -2)=0$
$\because cos\theta =0\Rightarrow \theta =\frac{\pi }{2}$ (Not possible)
so, we must have $3co{s}^2\theta -2=0$
$cos\theta =\sqrt{\frac{2}{3}}$ i.e


Hence $sin\theta =\frac{1}{\sqrt{3}}$
Height of cylinder $=2Rsin\theta =\frac{2R}{\sqrt{3}}$