What is the largest area that can be enclosed?

Question

You have 200 yards 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. Please show steps.

Answer 1

Let x = length of the side of the fence that is parallel to the river and let y  be the length of each of the other two sides.  

 

x + 2y = 200, so x = 200 - 2y

 

Area = xy = (200-2y)(y)

 

              = -2y²+200y

 

Maximum area when y = -200/(2(-2)) = 50 yards

                             x = 200 - 2y = 100 yards

 

 

Answer 2

Let y be the length of the side parallel to the river, x is the side perpendicular.

Since the side by the river has no fence, the total perimeter is: y + 2*x = 200 yards (given)

 

The area, A, is xy

from the equation for the perimeter, we have that y=200-2*x = 2*(100-x)

So A=x*2*(100-x) = 2*(100x - x^2)

 

The area is zero when x = 0 and when x = 100

So the maximum must be halfway between those when x = (100 + 0 )/2 = 50 yards

And therefore, the other dimension is: 200 - 2* 50 = 100 yards