Question

prove that 3πvh^3-c^2h^2+9v^2=0

Answer 1

c = π r L, where r = radius of the base of the cone and L = slanting length of the cone

=> c = π r * √(r^2 + h^2)

=> c^2 = (π r)^2 * (r^2 + h^2) ... (1)

Volume of the cone,

V = (1/3) π r^2 h

=> r^2 = (3V) / (π h)

Plugging this value of r^2 in (1),

c^2 = (π)^2 * (3V) / (π h) * [(3V) / (π h) + h^2]

=> c^2 = (9V^2) / h^2 + 3πVh

=> c^2h^2 = 9V^2 + 3πVh^3

=> 3πvh^3 - c^2h^2 + 9v^2 = 0.