Show that the height of a closed right circular cylinder of given surface and maximum volume, is equal to the diameter of its base.

Explanation

Let r be the radius of base and h be the height of cylinder.
Therefore, its total surface area
` S = 2 pi r h + 2 pi r^(2)` …(1)
If volume is V, then
` V = pi r^(2) h`
` = pi r^(2) ((S-2 pi r^(2))/(2pi r))` [From equation (1)]
` S/2 r - pi r^(3)`
` rArr (dV)/(dr) = S/2 - 3 pi r^(2)`
For maxima/minima ` (dV)/(dr) = 0`
` S/2 - 3 pi r^(2) = 0`
` rArr (2pi r h+2 pi r^(2))/2 - 3pi r^(2) = 0`
` rArr pi rh +pi r^(2) - 3 pi r^(2) = 0`
` rArr pi rh = 2 pi r^(2)`
` rArr h = 2r`
and ` (d^(2)V)/(dr^(2)) = 0 - 6 pi r =- 6 pi r lt 0 `
Therefore, volume is maximum at h = 2r.