STATEMENT OF PROBLEM

A machine that cuts pie shaped slices (sectors) from circular pieces of tin is set by entering the size of the angle to be cut in radians

[0, 2 Pi] . The resultant piece of tin is then welded together along the edges of the cuts to form a cone. The machine is now set to the appropriate angle to form cones of maximum volume from circular pieces 4 cm. in radius.

1) What is the current setting of the machine in radians and what is the volume of the resulting cone, i.e. what is the maximum volume cone we can obtain from this process?

2) A cylindrical tin can that presently holds exactly 12 fl. ounces (354 ml) has a height of 12 cm. with a flat top and base. If a cone of the above type (i.e. cut from circle of radius 4 cm) replaces the flat top, how much more volume will be added and what will be the size of the angle of the cut needed to construct it?