Mensuration-46

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If h, c and V be the height, curved surface and volume of a cone, show that
3 Π V h3 - c2 h2 + 9V2 = 0

Solution: Let r be the radius and l the slant height of the cone.

Then c = curved surface = Π r l = Π r √(h2 + r2)

V = Volume = 1/3 x Π r2 h

Given expression = 3 Π V h3 - c2 h2 + 9V2

= 3 Π (1/3 x Π r2 h) h3 - Π2 r2 (h2 + r2) h2

+ 9 (1/9 Π2 r4 h2)

= Π2 r2 h4 - Π2 r2 h4 -

Π2 r4 h2 + Π2 r4 h2 = 0 = R.H.S.

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