In answer to the Perimeter-Area Quiz from last week:
The area of a rectangle is always maximized, relative to perimeter, when the rectangle is a square.
To the left is an example where the perimeter of the rectangles remains a constant length of 24 units. The area increases as the sides of the rectangle get closer in length to each other.
To the right is an example where the area of the rectangles remains constant at 16 square units. The long thin rectangle has a perimeter of 34 units, the next rectangle has a smaller perimeter of 20 units, but the square's perimeter is just 16 units.
Both examples show that the maximum area of a rectangle, relative to its perimeter, occurs when the rectangle is a square.