Problem: Maximum Volume of a Cone
A circular disc of radius R is used to make a Cone
by removing a sector with angle ø and then joining the
edges where the sector was removed. What is the maximum volume
that such a cone can attain? That is, what angle 
for
a disc of radius R would you remove to make a cone of maximum
volume?
Suggestion: Make a physical model to demonstrate to yourself or to others how a cone is formed from a circular disc in this way. Cutting a circular disk out of the stock of a manila folder is about the right weight. Rather than completely removing the sector you are taking out, try cutting along a single radius. Then slide the edges over the top of one another to form a cone.
In this illustration the sector of the circular disc is removed
and the edges joined to form the cone as shown below where r is
the radius of the base of the cone. The circumference of the base
of the cone is 
This is a standard calculus problem. Can you find ways to investigate the problem without calculus?
Suggestion: Determine a function for the volume as the angle is varied. Make a graph of the function. Interpret the graph.
Suggestion: Try building a model of the problem with GSP.
Suggestion: Try building a spreadsheet with the volume determined for different angles.
Suggestion: Try using the AM-GM inequality. Why does it not work?
Do you need Help with setting up the algebra for this?