Math Help - Maximisation Problem - Cone/Sphere

Problem:
What is the volume of the largest sphere that can be placed inside a cone with a slant height of 10?


What I've done so far:

• Recognize that finding the largest radius will suufice; once I find it, I can find the volume easily.
• Reduce the problem to a right triangular cross-section with an inscribed semicircle. the semicircle has a diameter that lies on the height of the triangle. A radius has been drawn to the right angle vertex of the triangle, and to the hypotenuse of ten (at a right angle).

If I call the far angle across from the semicircle , then:

Or,

Past this, I am stumped. An Implicit derivative doesn't seem to get me anywhere, but it is impossible to fully separate my two variables. Any thoughts?

let be the base angle of the cone.

Note that a line from the center of the circle to the vertex of bisects .








find and maximize ...

I get the value