A farmer wishes to enclose a pasture that is bordered on one side by a river (so one of the four sides won't require fencing).

Question

She has decided to create a rectangular shape for the area, and will use barbed wire to create the enclosure. There are 600 feet of wire available for this project, and she will use all the wire. What is the maximum area that can be enclosed by the fence? (Hint: Use this information to create a quadratic function for the area enclosed by the fence, then find the maximum if the function.)

Answer 1

Let x = length of fenced side parallel to the side that borders the river

     y = length of each of the other two fenced sides

 

Then, x + 2y = 600

 

           So, x = 600 - 2y

 

Area = xy = y(600-2y)

 

               = -2y² + 600y

 

The graph of the area function is a parabola opening downward.

 

  The maximum area occurs when y = -600/[2(-2)] = 150

                                               x = 600-2y = 300

 

To maximize the area, the fenced side parallel to the river should be 300 feet long, while each of the other two fenced sides should be 150 feet long.   

Answer 2

w is the width of the pen

600 - 2w is the length

 

A = w(600 - w)

A = 600w - w²

A = -w² + 600w

 

Can you determine the zeros?