Question

If h.c and v are height, the curved surface area of the volume of cone respectively, prove 3πrh2-c2h2+9v2=0.

Answer 1

Let, h,c and v are the height, the curved surface area and volume of the cone respectively.
∴, c=πrl where r is the radius and l is the slant height of the cone.
Also, l²=h²+r²
v=1/3πr²h
∴, 3πvh³-c²h²+9v²
=3π×1/3πr²h×h³-π²r²l²h²+9×1/9π²r⁴h²
=π²r²h⁴-π²r²l²h²+π²r⁴h²
=π²r²h⁴-π²r²(r²+h²)h²+π²r⁴h²
=π²r²h⁴-π²r⁴h²-π²r²h⁴+π²r⁴h²
=0 (Proved)
[**LHS should be 3πvh³-c²h²+9v²]