A Gardener has 560 feet of fencing to fence in a rectangular garden. One side of the garden is bordered by a river so it does not need any fencing. What dimensions would guarantee that The Gardener has the greatest possible area

The dimensions that give the largest possible area are 186 by 187, with an area of 34,782ft².

To maximize area and minimize perimeter, you make the figure as close to equilateral as possible. 560/3 = 186 2/3. We would rather use whole number dimensions, so we would have 186 for one side. 560-186 = 374. 374/2 = 187 for the other dimension.