A cone is made from a circular sheet of radius R by cutting out a sector and gluing the cut edges of the remaining piece together (see figure). What is the maximum volume attainable for the cone? Solution

A rectangular plot of land is to be fenced in using two kinds of fencing. Two opposite sides will use heavy-duty fencing selling for $3 a foot, while the remaining two sides will use standard fencing sellling for $2 a foot. What are the dimensions of the rectangular plot of greatest area that can be fenced in at a cost of $6000? Solution

Prove that (0,1) is the closest point on the unit circle to (0,2). Solution