I've tried and tried, and visualized, but I can't find the relationship between the cone's radius/height and that of the cylinder. If someone could help me with that I'll be fine for the rest.
Determine the maximum volume of a cylinder that can be inscribed in a cone whose height is 3 cm and whose base radius is 3 cm.
The cylinder will have volume 4/9th that of the cone. Let's do a general case and then you sub in your values.
Let R and H be the radius and height of the cone, and r and h be the radius and height of the cylinder.
The volume of the cylinder is
Now, we use similar triangles.
Solve for h:
Sub into V:
So, the max volume is
Use your values now.