If h, c and v be the height, curved surface and volume of a cone, show that 3πvh^3 - c^2h^2 + 9v^2 = 0.
If h, c and v be the height, curved surface and volume of a cone, show that 3πvh3 - c2h2 + 9v2 = 0.
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Let the radius of the base and slant height of the cone be r and l respectively. The n;
c = curved surface = πrl = πr√(r2 + h2) ....(i)
v = volume = 1/3πr2h ...(ii)
∴ 3πvh3 - c2h2 + 9v2 = 3π(1/3π2r2h)h2 - π2r2(r2 + h2)h2 + 9(1/3πr2h)2 [Using (i) and (ii)]
= 2r2h4 − π2r2h2 − π2r2h4 + π2r4h2 = 0.
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