A right cylinder is inscribed in a sphere of radius r. How do you find the largest possible volume of such a cylinder?

Question

asked Mar 7, 2021 90.8k views

1 Answer

TheKvist · Answer 1 · 2021-03-12T15:20:33+0000

Answer:

Explanation:

Suppose Radius of sphere is R and cylinder inscribed inside the sphere is r with height h

using Pythagoras theorem in shown triangle

R^2=r^2+((h)/(2))^2

r^2=R^2-((h)/(2))^2

Volume of cylinder is

$V=\pi * r^2* h$

$V=\pi * (R^2-((h)/(2))^2)* h$

$V=\pi (R^2\cdot h-(h^3)/(4))$

differentiate V w.r.t to h we get

$\frac{\mathrm{d} V}{\mathrm{d} h}=\pi (R^2-(3h^2)/(4))$

Putting
$\frac{\mathrm{d} V}{\mathrm{d} h}$ to get maxima/minima

R^2=(3h^2)/(4)

h=(2R)/(√(3))

therefore radius r is given by

$r=\sqrt{(2R)/(3)}$

therefore volume is given by

$V=(4)/(3√(3))\pi R^3$