Prove that a conical tent of given capacity will require the least amount of canvas when the height is √2 times the radius of the base.

Question

Philoid · Accepted Answer

Let the radius and height of cone be r and h respectively

It is given that volume of cone is fixed.

Volume of cone, V =

h = …1

Curved surface area of cone, S = πrl (l is slant height)

Since,

So,

Condition for maxima and minima is

…2

For , > 0

S will be minimum for

From equation 1

(from equation 3)

h = √2 r