Question

A right circular cylinder which is open at the top and has a given surface area, will have the greatest volume if its height h and radius r are related by

Options

• 2h = r

• h = 4r

• h = 2r

• h = r

Answer 1

h = r

Explanation:

Volume of cylinder, (V) = `pir^2h`

Surface area, (S) = `2pirh + pir^2` .......(i)

⇒ h = `(S - pir^2)/(2pir)`

∴ V = `pir^2 [(S - pir^2)/(2pir)]`

= `r/2[S - pir^2]`

= `1/2[Sr - pir^3]`

Now, Differentiate both sides, w.r.t 'r'

`(dV)/(dr) = 1/2[S - 3pir^2]`

Now, circular cylinder will have the greatest volume, when `(dV)/(dr)` = 0

⇒ S = 3pr²

⇒ `2pirh + pir^2 = 3pir^2`

⇒ `2pirh = 2pir^2`

⇒ r = h.