1. Prove that, among all rectangles with fixed perimeter p, where p > 0, the largest in area is a square.

2. A 2 × 3 array of six congruent rectangular pigpens (that all look the same from above) will be in the overall shape of a rectangle R. We may use 100 feet of fencing to form the boundaries of the pigpens. Find the dimensions for a single pigpen that will maximize the total area of all the pigpens, and find this total area. (The fencing separating the pigpens has constant height, so we may ignore height in our calculations. Also, assume the boundaries between pigpens are not double-fenced; that is, assume that the thickness of the fencing between pigpens is the same as the thickness of the fencing along the outer boundary, R.)

3. A glass aquarium is to be shaped as a right circular cylinder with an open top and a capacity of two cubic meters. Find the dimensions of the valid cylinder that requires the least amount of glass, and find that amount of glass. (Ignore the thickness of the glass.) How would the diameter compare to the height?