You have a total of 860 feet of fencing to enclose a large r

You have a total of 860 feet of fencing to enclose a large rectangular area and divide it into four smaller pens of the same dimensions. The fencing used to divide the large pen must be parallel to the same side of the large rectangle, as shown below. Your goal is to maximize the total rectangular area. Give the dimensions as well as the maximum area. Show your steps leading up to the final answer

Solution

Let L = length and W = width of rectangle

Now width W is used 5 times to make pens

So perimeter = 2L+5W = 860

=> L =(860-5W)/2

Also Area = WL = W(860-5W)/2 = (860W-5W²)/2

A\' = (860-10W)/2 = 0

=> 430-5W=0

=> 430=5W

=> 430/5 = W

=> W = 86

Also L = (860-5*86)/2 =430/2 = 215

Now also A\" = -10/2 = -5 <0

Hence Area is maximum for L = 215 and W=86

The largest area is thus given by A = 86*215 = 18490 ft²