Question

Find the dimensions of a rectangular corral split into 2 pens of the same size producing the greatest possible enclosed area given 100 feet of fencing

length=

width=

Answer 1

Let the length and the width of the rectangular corral split into 2 pens of the same size be x ft. and y ft. respectively. If 100 feet of fencing suffices, then new have 2x +3y = 100…(1). Also the area of the corral is A(x,y) = xy = x(100-2x)/3 = 100x/3 -2x²/3.

Now, we know that A(x,y) is maximum when dA/dx = 0 and also d² A/dx² is negative. Here, dA/dx = 100/3 – 4x/3 . Thus, if dA/dx =0 , then 4x/3 = 100/3 so that x = 25. Further, d² A/dx² = -4/3 which is always negative regardless of the value of x. Thus, the area of rectangular corral split into 2 pens of the same size i.e A9x,y) is maximumwhen its length is 25 ft. Then its width, i.e. y = (100-2*25)/3 = (100-50)/3 = 50/3 ft.

Thus, for maximum area,

length= 25 ft.

width=50/3 ft.