Lucy bought 162 feet of fencing with which to enclose a rectangular garden. of the lengths of the garden's sides are whole numbers, what is the maximum number of square feet she can enclose with the fencing?

The rectangle that uses it's perimeter to enclose the most area is a square.

If the perimeter is 162 ft, then each side of the square is

162 / 4 = 40.5 ft.

The area is

(40.5) x (40.5) = 1,640.25 square feet.

If the sides have to be whole numbers, then the closest she can get is 40 ft x 41 ft.
The area is 1,640 square feet.