Question

For a right circular cone inscribed in a sphere of radius 'r', find the ratio of the altitude to the base radius of the cone of the largest possible volume.

(A) 1:√2
(B) 1:√3
(C) 1:2
(D) 2:1

Answer 1

The ratio of the altitude to the base radius of a cone of the largest possible volume inscribed in a sphere is 1:√3, option (B), determined using calculus and the Pythagorean theorem in the context of the cone and sphere dimensions.

Step-by-step explanation:

To find the ratio of the altitude to the base radius of a cone of the largest possible volume inscribed in a sphere of radius 'r', we can use calculus to maximize the volume function of the cone. The formula for the volume of a cone is V = (1/3)πr2h, where r is the base radius and h is the altitude. However, for a cone inscribed in a sphere, the altitude h, the base radius r, and the sphere's radius 'r' form a right-angled triangle where 'r' is the hypotenuse. Therefore, we have the relation 'r'2 = h2 + r2. Solving this for maximum volume yields the answer (B) 1:√3 as the ratio of the altitude to the base radius of the cone with the largest possible volume.