What is the maximum area that can be enclosed by the pen?

Question

A farmer wants to create a rectangular pen for cows to graze. She has 1500m of fence and wants to have the pen border on a creek so that the cows will have water and she will have to fence only three sides of the pen. What is the maximum area that can be enclosed by the pen? 

Answer 1

Let length = x

Let width = y

 

We can create equations using these variables.

 

Since she only needs to fence 3 sides of the pen, her perimeter is then

 

2x + y = 1500         or        x + 2y = 1500             eq1

 

Area = xy                                                             eq2

 

 

Next, we substitute eq1 into eq2.  We can have eq2 in terms of x or in terms of y, so that we can have a quadratic function for the area.

 

Area = x(1500 - 2x)          or           Area = y(1500 - 2y)

 

Area = -2x² + 1500x                       Area = -2y² + 1500y

 

 

As you can see, either arrangement gets you the same function of area, just different use of variables.  However, lets pick the function in terms of x, being that all standard functions are in terms of x.

 

 

To find the maximum area, we need to find the vertex of this quadratic function.

 

Vertex has coordinate (h, k).

 

h = -b / (2a)

 

where:

a = -2

b = 1500

 

 

Plug in these values to find the value of h.  This will be the x-coordinate of the vertex.

 

h = -1500 / (2*-2)

h = -1500 / -4

h = 375

 

Evaluate the area when x=375.

 

Area = -2(375)² + 1500(375)

Area = 281250

 

Maximum area is 281250 m².