Question 91638
Let's start with the appropriate formulas for the volumes of the given solids.
Cylinder:
{{{V[c] = (pi)r^2h}}} but the height of the cylinder is given as the length of the diameter (D) of its base and D = 2r where r is the radius of the base. So we can express the volume of the cylinder entirely in terms of the radius (r) of its base, right?
{{{V[c] = (pi)r^2(2r)}}} Simplifying this we get:
{{{V[c] = 2(pi)r^3}}} as the volume of the cylinder.
Sphere: 
The largest sphere that could be contained within a cylinder of the dimensions given above i.e., h = 2r and radius r, would be a sphere whose radius is equal to that of the cylinder, or radius r.
The volume of a sphere of radius r is given by:
{{{V[s] = (4/3)(pi)r^3}}}
Now all we have to do is to subtract the volume of the sphere from the volume of the cylinder, or {{{V[c]-V[s]}}} to find the amount of material that must be removed from the wooden cylinder to create the largest possible sphere.
{{{V[c]-V[s] = 2(pi)r^3 - (4/3)(pi)r^3}}} = {{{(2/3)(pi)r^3}}}