A cone with semi-vertical angle 60° is inscribed in a sphere. What is the ratio of the volume of the cone to that of the sphere?
Correct: C
Let the radius of the sphere be R and the radius of the base of the cone be r. From the geometry, we get r = R cos 60° = R/2. The height of the cone is h = R sin 60° = R√3/2. The volume of the cone is Vc = 1/3 πr^2h = 1/3 π(R/2)^2(R√3/2) = πR^3√3/24. The volume of the sphere is Vs = 4/3 πR^3. Therefore, the ratio Vc/Vs = πR^3√3/24 divided by 4/3 πR^3 = √3/32.