# Prove that the se 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

From(1)

2r6 = (r2h)2

2r6 = r4h2

2r2 = h2

Since height and radius cannot be negative, …… (5)

From the figure From(5)  Thus proved.

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