1. ## Optimization

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
r^2 + z^2 = a^2
so that
z^2 = a^2 - r^2
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)πr^2h

Sub in h
V = (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.

2. Originally Posted by mortalcyrax
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
r^2 + z^2 = a^2
so that
z^2 = a^2 - r^2
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)πr^2h

Sub in h
V = (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.
Use the intersecting chord theorem to relate the radius of the base of the cone to its height and the radius of the sphere. Then substitute either the radius of the base of the height of the cone from that relation into the formula for the volume of the cone. Then find the minimum volume of the cone.

CB

3. Problem has been solved