Question

the height, curved surface area and volume of a cone are H, C and V respectively Prove that: 3pie v h sq - c sq h sq + 9v sq =0

Answer 1

The curved surface area of a cone 

=C

=πrl

=πr√(h²+r²)

where h, l and r are the height, slant height and radius of the cone.

Volume of the cone

=V

=πr²h/3

∴3πVh³-C²h²+9V²

=3π{(πr²h)/3}h³-{πr√(h²+r²)}²h²+9(πr²h/3)²

=π²r²h⁴-π²r²(h²+r²)h²+9π²r⁴h²/9

=π²r²h⁴-π²r²h⁴-π²r⁴h²+π²r⁴h²

=0 (Proved)