Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius `R` is `(2R)/(sqrt(3))` .

Explanation

Let radius be `r ` and height be `h` of the inscribed cylinder.

`implies h=2sqrt{R^2-r^2}`

Volume of the cylinder `=pir^2h=2pir^2sqrt{R^2-r^2}`

`{dV}/{dr}={4pirR^2-6pir^3}/sqrt{R^2-r^2}`

Put `{dV}/{dr}=0`

Thus `{4pirR^2-6pir^3}/sqrt{R^2-r^2}=0 `

`implies r^2=2/3R^2`

And `{d^2V}/{dr^2}={4piR^4-22pir^2R^2+12pir^4+4pir^2R^2}/(R^2-r^2)^{3/2}`

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