Prove that the semi - vertical angle of the right circular cone of given volume and least curved surface is .
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:
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Let us assume r2h = k(constant) …… (1)
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⇒ …… (2)
We know that surface area of a cone is
⇒ …… (3)
From the cross - section of cone we see that,
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⇒ …… (4)
Substituting (4) in (3), we get
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From (2)
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⇒
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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
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Differentiating using U/V rule
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⇒
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Equating the differentiate to zero to get the relation between h and r.
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⇒
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
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From(5)
⇒
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∴ Thus proved.