You have 200 feet of fencing to enclose a rectangular plot that borders on a river. If you do not fence the side along the river, find the length and width of the plot that will maximize the area. What is the largest area that can be enclosed?
Let length of plot be “l” feet and breadth be “b” feet. Suppose edge of river is along a length of the plot.
So, fencing used is l+b+b (two breadths sides and one length side- opposite to the river side)
Thus l+2b = 200. Thus l = 200-2b.
Also area of plot is l*b feet squared.
So, area = (200-2b)b
To find max area we differentiate it with respect to “b” and equate it to zero.
d (200b-2b2) / d (b) = 0
Hence, 200 - 4b = 0.
So, b = 50 feet and l = 200 - 2b = 100feet
Max area = lb = 5000 feet squared.