A sector with central angle is cut from a circle of radius 12 inches, and the edges of the sector are brought together to form a cone. Find the magnitude of so that the volume of the cone is a maximum.
See attached figure
- The radius of the sector which is the slanted line from the tip of the cone to it's edge
- the radius R of the base circle of the cone.
You know the length l of the arc of the sector:
which has the same length as the circumference of the base circle:
The slanted line s, the radius of the base cicle R and the height of the cone form a right triangle. Use Pythagorean theorem:
Plug in the terms of R and h into the equation of the volume. You'll get a function with respect of :
To calculate the maximum volume you have to derive this function and solve for the equation:
EDIT: Removed my typo in the last equation.