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 thatr^2 + z^2 = a^2
orz^2 = a^2 - r^2
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)πr^2h
Sub in hV = (1/3)πr^2(a + sqrt(a^2 - r^2))
This is where I get stuck because I do not know where to go from here.