Question

Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is 8/7of the volume of the sphere.

Answer 1

HELLO DEAR,

let the radius of the given sphere be a.
let V be the volume of the inscribed cone,
h be it's height and r be it's radius.

then V = 1/3πr²h.

NOW, OD = (AD - OA) = (h - a).

OC² = OD² + DC²

a² = (h - a)² + r²

r² = h(2a - h).

therefore, V = 1/3πh²(2a - h)

= [2πah²/3 - 1/3πh²].

therefore,dV/dh = (4πah/3 - πh²)

and d²V/dh² = (4πa/3 - 2πh).

so, dV/dh = 0

h = 4a/3 and [d²V/dh² at h = 4a/3] = -4πa/3 < 0.

therefore, V is maximum when h = 4a/3 and r² = 8a²/9.

maximum volume = 1/3π×8a²/9 × 4a/3
= 8/27×(4/3πa³).

hence, that the volume of the largest cone that can be inscribed in a sphere of radius R is 8/27of the volume of the sphere.

I HOPE ITS HELP YOU DEAR,
THANKS