Finding the ratio of 2 radii

Question

Suppose you drill a circular hole with radius r through the center of a sphere with radius R. You remove exactly half the volume of the sphere. The ratio of your radii is r/R.  What is the ratio?

Answer 1

It sounds like they want you to assume the drilled out part is a cylinder (even though it's not quite since the top and bottom will be rounded).  The volume of the cylinder you drill out is V₁=(1.3)*pi*r²*h.  The volume of the sphere is V₂=(4/3)*pi*R³.  Since you've drilled out half of the sphere's volume we can set up the following equation:

 

V₁=.5*V₂

or

(1.3)*pi*r²*h = .5*(4/3)*pi*R³

 

The only other thing you need to worry about before solving for r/R is h.  Since the hole goes through the entire sphere it will be equal to 2R.  Plug that into the equation and then solve for r/R.

 

(Upon rereading the question, it occurs to me that this may be a higher level calculus problem where you would take the curvature into account.  If that is so, then you can probably use my answer as a guideline to help you solve the problem, but it won't lead to an actual solution without using calculus.  In that case you will need to split the drilled part into 3 pieces: top and bottom and middle.  The top and bottom being identical discs with a rounded top, and the middle being a cylinder.)