Question

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

Answer 1

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`