What is the maximum area of the pen?

Question

A farmer want to create two equal pens along the side of his barn as shown in the diagram below. The barn side needs no fencing. If he has 3600 m of fencing, find the maximum area of each pen.

Answer 1

I am still learning, but still I wanted to try it, so..

Let x and y be length and breadth of the barn.

Given in figure and your description, x + 3y = 3600 => y = (3600 - x)/3,

Therefore, Area occupied by the fence, A = x * y = x * (3600 - x)/3= (3600x - x²)/3

So the max area is when dA/dx = d/dx {(3600x - x²)/3}  =0 => x = 1800

so, x =1800 and y = (3600 -1800)/3 = 600

So, Max area A = x* y = 1800 * 600 = 1080000

Since two equal pens must be created, Therefore the area of a single pen = 1080000/2 = 540000 sq. unit

Answer 2

We can let L=length of the barn, which is also the length of fencing parallel to the barn, and we can use W as the perpendicular distance of this length of fencing from the barn, which is also the width of each pen. The length of each pen is L/2 because the pens are the same size.

The amount of fencing is L+3W=3600m. The area of each pen is LW/2, and the total area A of the pens is A=LW=(3600-3W)W, substituting for L. So A=3600W-3W²=-3(-1200W+W²). Complete the square in brackets: A=-3(W²-1200W+360000-360000)=-3(W-600)²+1080000. When W=600m, we can see that the maximum area is 1080000m², because all other values of W reduce 1080000 by some amount. Therefore L=3600-3W=3600-1800=1800m. The size of each pen is 900 by 600, area 540000m².