SOLUTION: Peter’s farm has 160 meters of fencing, and he wants to fence a rectangular field that borders a straight river. He needs no fence along the river side. Find the largest area of

Since one side is the river, the rectangle's fence perimeter will be

L + 2W = 160.

Hence, L = 160 - 2W.

Area = Length * Width.

Substitute (160-2W) for L:

A = W(160 - 2W)

A = -2W^2 + 160W.

This is a quadratic function. It has the maximum at x = -b/(2a), according to the general theory.

    (See the lessons
     
         - HOW TO complete the square to find the minimum/maximum of a quadratic function

         - Briefly on finding the minimum/maximum of a quadratic function

     in this site).


For our quadratic function the maximum is at

W =  =  = 40.

So, W = 40 meters is the width for max area.


Then the length is  L = 160 - 2W = 160 - 2*40 = 80 meters.


Then the maximal area is L*W = 80*40 = 3200 square meters.


The plot of the quadratic function  y = - 2x^2 + 160x  for the area is shown below:  y = area and x = width.



     


    Plot y = -2x^2 + 160x.