Calculus Application problem

Question

An ecologist needs to fence in a state park with a rectangular plot. The rectangular plot must have an additional section of fencing across its width. Find the maximum area that can be enclosed with 600 feet of fencing.

Answer 1

Let length = x

Let width = y

 

Area = xy

 

The perimeter equation is

 

2x + 2y + x = 600

 

The extra x dimension represents the additional section of fencing.

 

3x + 2y = 600

 

 

We can solve for y from this equation.

 

2y = -3x + 600

 

   y = (-3/2)x + 300

 

 

Substitute this value of y into the area formula.

 

Area = x((-3/2)x + 300)

Area = (-3/2)x² + 300x

 

Take the derivative of area and set it equal to zero.  Then solve for x.  The value of x is then your length that will give the maximum area.

 

-3x + 300 = 0

 

-3(x - 100) = 0

 

 

Then length must be 100 feet.  Plug in this value into the area to get maximum area.