Question

A right circular cylinder is to be made so that the sum of the radius and height is 6 metres. Find the maximum volume of the cylinder.

Answer 1

Let r be the radius of the base and h be the height of cylinder.

Given: r + h = 6

∴ h = 6 – r   ....(1)

Volume, V = πr²h

i.e. V = πr²(6 – r)   ...[From (1)]

i.e. V = π(6r² – r³)

∴ `(dv)/(dr)` = π(12r – 3r²)

Put `(dv)/(dr)` = 0

`\implies` π(12r – 3r²) = 0

∴ 12r – 3r² = 0

`\implies` 3r² = 12r,

`\implies` r = 4

∴ h = 2   ...[From (1)]

Also `(d^2v)/(dr^2)` =  π(12r – 6r)

∴ `((d^2v)/(dr^2))_(r = 4)` = π(12 – 24)

= –12π < 0

`\implies` V is maximum when r = 4 and h = 2 metres.

∴ Maximum volume = πr²h

= π × 16 × 2

= 32π cu. meters.