Answer:
Step-by-step explanation:
Suppose Radius of sphere is R and cylinder inscribed inside the sphere is r with height h
using Pythagoras theorem in shown triangle
[tex]R^2=r^2+(\frac{h}{2})^2[/tex]
[tex]r^2=R^2-(\frac{h}{2})^2[/tex]
Volume of cylinder is
[tex]V=\pi \times r^2\times h[/tex]
[tex]V=\pi \times (R^2-(\frac{h}{2})^2)\times h[/tex]
[tex]V=\pi (R^2\cdot h-\frac{h^3}{4})[/tex]
differentiate V w.r.t to h we get
[tex]\frac{\mathrm{d} V}{\mathrm{d} h}=\pi (R^2-\frac{3h^2}{4})[/tex]
Putting [tex]\frac{\mathrm{d} V}{\mathrm{d} h}[/tex] to get maxima/minima
[tex]R^2=\frac{3h^2}{4}[/tex]
[tex]h=\frac{2R}{\sqrt{3}}[/tex]
therefore radius r is given by
[tex]r=\sqrt{\frac{2R}{3}}[/tex]
therefore volume is given by
[tex]V=\frac{4}{3\sqrt{3}}\pi R^3[/tex]
