if h,c,v are respectively the height, curved surface area and the volume of a cone.prove that

3 pie v h cube - c2h2 + 9v2 =0

Question

if h,c,v are respectively the height, curved surface area and
the volume of a cone prove that
3 pie v h cube - c2h2 + 9v2 =0

Lalit Mehra · Accepted Answer

Given h, C, V are the height, curved surface area and volume of the cone respectively.

∴ C = πrl and $V = \frac{1}{3} π r^{2} h$

Slant height of the cone, $l = \sqrt{r^{2} + h^{2}}$

3πVh³ – C²h² + 9V²

$= 3 π (\frac{1}{3} π r^{2} h) h^{3} - π^{2} r^{2} l^{2} \times h^{2} + 9 (\frac{1}{3} π r^{2} h)^{2}$

= π² r²h⁴– π² r²(r²+ h²) h²+ π²r⁴h²

= π² r²h⁴– π² r⁴h²– π²r²h⁴+ π²r⁴h²

= 0

Thus, 3πVh³ – C²h²+ 9V² = 0