SOLUTION: A rectangular plot of land is to be enclosed by fencing. One side is along a river and so needs no fence. If the total fencing available is 1400 meters, find the dimensions of the

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

L + 2W = 1400.

Hence, L = 1400 - 2W.

Area = Length * Width.

Substitute (1400-2W) for L:

A = W(1400 - 2W)

A = -2W^2 + 1400W.

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 =  =  = 350.

So, W = 350 ft is the width for max area.

Then the length is  L = 1400 - 2W = 1400 - 2*350 = 700 ft.


Find the max area. It is

A = L*W = 700*350 = 245000 square feet.


The plot of the quadratic function for the area is shown below:  y = area and x = width.