A farmer has 2200 feet of fencing available to enclose a rec

A farmer has 2200 feet of fencing available to enclose a rectangular area bordering a river. If no fencing is required along the river, find the dimensions of the fence that will maximize the are. What is the maximum area? Find the dimensions of the fence that will maximize the area Width = feet Length = feet The maximum area is square feet.

Solution

Let the length and the width of the plot be x ft and y ft respectively. Then, since no fencing is required along the river ( which we presume to be the length), we have x + 2y = 2200, or, y = ½( 2200 –x). The area A of the rectangular plot is A = length * width = xy = ½ [ x * (2200 –x) ]= 1100x – (½)x² . Then dA/dx = 1100 –x and d²A/ dx² = -1. For maximizing A, we must have dA/dx = 0 and also d²A/dx² should be negative. Now dA/ dx = 0 when 1100 –x = 0, i.e. when x = 1100. Also, d²A/dx² is always negative regardless of the value of x so that x = 1100 will give us the maximum area. Further, if x = 1100, then y = ½( 2200 –x) = ½ ( 2200 -1100) = 1100/2 = 550. Thus, the length and the width of the plot are 1100ft and 55o ft. respectively, the length being along the river.