Hello, lauren72!

Did you make a sketch?

A cylinder is inscribed in a right circular cone of height 4 and base radius 4.5.

What are the dimensions of such a cylinder which has maximum volume? Look at a cross-section of the cone. Code:

- *
: * | *
: * | *
: * | *
4 * - - - + - - - *
: * | | | *
: * | |h | *
: * | | | *
- *-------+-------+-------+-------*
: 4.5 : r : 4.5-r :

The radius of the cylinder is , its height is

From the similar right triangles: . .[1]

The volume of a cylinder is: . .[2]

Substitute [1] into [2]: .

Differentiate and equate to zero: .

Then: .

Hence: .

Substitute into [1]: .

Therefore: .