Hello, Philip!

2. Determine the dimensions of the cone of maximum volume

that can be inscribed in a sphere with radius 3 Code:

A
* * *
* /:\ *
* / : \ *
* / :3 \ *
/ : \
* / : \ *
* / O* \ *
* / * : * 3 \ *
/ * y: * \
B *--------+--------* C
* x *
* *
* * *

The center of the sphere is

Note that

is the side view of the cone.

The volume of a cone is: .

In the diagram, the radius is and the height is

So we have: . **[1]**

The diagram also says: . **[2]**

Substitute **[2]** into **[1]**: .

We have: .

Then: .

We have the quadratic: .

. . which factors: .

. . and has roots: .

Since the measurements are positive,

. . and from **[2]**: .

Therefore, the cone of maximum volume has radius and height