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

Prove that a conical tent of given capacity will require the least amount of canvas when the height is \(\sqrt2\) times the radius of the base.

Answer 1

a

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 = \(\frac{1}{3}\pi r^2h\)

⇒ h = \(\frac{3V}{\pi r^2}\) ...(1)

Curved surface area of cone, 

S = πrl (l is slant height)

Since,

⇒ h = √2r