Question

If the cylinder of largest possible volume is inscribed in a given sphere, the ratio of the volume of the sphere to that of the cylinder is what?

Answer 1

the ratio would be 23

Answer 2

its a multiple choice question and it could be
sqrt(3) : 1
sqrt(3) : 3
3 : 1
2sqrt(3) : 3
3sqrt(3) : 4

Answer 3

I got 4sqrt(2):3. I can't figure out where the sqrt(3) comes from.
Draw the sphere with the circle cylinder inside. Let r be the radius of the circle. Let a be the constant radius of the sphere and h/2 the height of the top half of the cylinder. You can use the pythagorean theorem to find the relationship between h/2 and r, and hence, h and r. You get h = 2sqrt(a^2 - r^2). Plug that into the Vc = pi r^2 h and maximize. Then find the ratio of Vs and Vc. But, clearly I'm making an error someplace.