Q. 21 4.0( 3 Votes )

Prove that the semi - vertical angle of the right circular cone of given volume and least curved surface is .

Answer :

Let ‘r’ be the radius of the base circle of the cone and ‘l’ be the slant length and ‘h’ be the height of the cone:

Let us assume ‘’ be the semi - vertical angle of the cone.

We know that Volume of a right circular cone is given by:

Let us assume r2h = k(constant) …… (1)

…… (2)

We know that surface area of a cone is

…… (3)

From the cross - section of cone we see that,

…… (4)

Substituting (4) in (3), we get

From (2)

Let us consider S as a function of R and We find the value of ‘r’ for its extremum,

Differentiating S w.r.t r we get

Differentiating using U/V rule

Equating the differentiate to zero to get the relation between h and r.

Since the remainder is greater than zero only the remainder gets equal to zero

2r6 = k2


2r6 = (r2h)2

2r6 = r4h2

2r2 = h2

Since height and radius cannot be negative,

…… (5)

From the figure


Thus proved.

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