Show that the right circular cone of least curved surface area and given volume has an altitude equal to √2 times the radius of the base.

Let the given volume be V and radius of base of cone be r and h be the height and l be the slant height of cone

Let S be the surface area

We have to find h such that the surface area is minimum so we have to establish a equation for S in terms of h

The surface area is given by S = πrl

Where l is slant height of cone given by

We need to eliminate r

For that given is volume of cone V

Square equation (a)

⇒ S² = π²r²(r² + h²)

Put value of r²

Now to find for what value of h S is minimum we first need to find the critical points such that

Differentiate S with respect to h and equate to 0

⇒ 0 = -18V² + 3Vh³π

⇒ 0 = -6V + h³π

⇒ h³π = 6V

Now we have to check whether is point of minima or maxima

If then h is a point of minima

Let us check

From (i)

Differentiate again with respect to h

The LHS will be differentiated using the uv rule which states that (uv)’ = u’v + uv’ where u = 2S and

Substitute value of from (i)

Put

Now observe that S represents surface area V represents volume which are measures for some quantity hence both are positive π is a positive constant hence and hence is a point of minima

Now let us find h in terms of r by putting the value of volume

Since

⇒ h² = 2r²

⇒ h = √2r

Hence for least curved surface area the altitude h has to be √2 times the base radius