Question

Show that the maximum volume of the cylinder which can be inscribed in a sphere of radius $5\sqrt{3}$ cm is $500\pi {cm}^3$ ?

Answer 1

$h^2+r^2=R^2=(5\sqrt{3}{)}^2=25\times 3=75$

$v=2\pi r^{2}h$

$v=kh(75-h^2)=k75h-h^3$

$Atmaximumv,$

$\frac{dv}{dh}=k75–3h^2=0$

$h^2=25\to h=5$

$r^2=75-h^2=75–25=50$

$maximumV=2\pi r^{2}h$

$=2\pi \left(50\right)\left(5\right)$

$=500\pi$