Question

The ratio of the altitude of the cone of greatest volume which can be inscribed in a given sphere to the diameter of the sphere is –

• A. 2/3
• B. ½
• C. 4/5
• D. 1/3

Answer 1

Answer: (A)

2/3

Answer 2

Let h be the height of the cone and r be its radius.

\ h = CL = CO + OL = a + OL

\ OL = h – a

r = LA = Ö(OA² – OL²)

or r = Ö{a² – (h – a)²} = $\sqrt{2ah - h^{2}}$

V = $\frac{1}{3}$pr²h = $\frac{1}{3}$p(2ah – h²) h

= $\frac{1}{3}$p (2ah² – h³)

dV/dh = (p/3) (4ah – 3h²) = 0

\ h = 0 or 4a/3

h = 0 is rejected Ž h = 4a/3 = (2/3) (2a)

h = $\frac{2}{3}$ (diameter).