Hello, Neverquit!

This is an intricate (and badly worded) problem . . .

Find the sector angle (where cone volume is maximized)

when a sector is removed and the remainder is joined to form a cone. Code:

* * *
* *
* *
* *
* O *
* o *
* * θ * R *
* *
* * * *
A o o B
* *
* * *
Rθ

The sector angle is

The radius of the circle is

The arc length of is: .

This is the circumference of the circular base of our cone.

. . .[1]

A cross-section of our cone looks like this:

Code:

*
/|\
/ | \
/ h| \ R
/ | \
/ | \
* - - + - - *
r

We have: .

. . Hence: . .[2]

The volume of a cone is: .

Substitute [1] and [2]: .

. . Therefore: .

And that is the function we must maximize.