**A cone is cut from a sphere of radius a. You are to find the cone with the largest volume. Calculate the volume of this cone exactly.**
Let variable

*r* be the radius of the circular base and variable

*h* the height of the inscribed cone as shown in the two-dimensional side view.

It

is given that the circle's radius is a. Let variable *z* be the height of the small right triangle.

By the Pythagorean Theorem it follows that

so that

or

z=sqrt(a^2 -

*r*^2)

Thus the height of the inscribed cone is

h = a + z = a + sqrt(a^2 - *r*^2)

We wish to MINIMIZE the total VOLUME of the CONE

V = (1/3)π*r*^2h

Sub in h

V = (1/3)π

*r*^2

(a + sqrt(a^2 -

*r*^2)

)

This is where I get stuck because I do not know where to go from here.