Expert: Sherry Wallin - 10/20/2009

Question
A rancher with 600 ft of fencing wants to enclose a rectangular area and then divide it into three pens with fencing parallel to one side of the rectangle.

a) Find a function that models the total area of the three pens.

b)Find the largest possible total area of the three pens.

Answer
Hi Suzanne~
    Let x = the length of the pen and y =  the width of the pen. If the plot is divided into 3 pens with fencing that is parallel to one side of the rectangle that means that there are two extra sides. So let it be that the side we call x has the two extra lengths of the side so 4x + 2y = 600. This means that the width y is [600 -4x]/2 = 300-2x. The area to maximize is then
[300-2x]x = -2x^2+300x. You can either use calculus and find the derivative of this function or realize this is a parabola that opens down so it has a maximum at its vertex which has x coordinate -b/2a = -300/(2*(-2)) = 75 and put that value of x back into the perimeter equation to find y:
4*75 + 2y = 600 -> 2y = 300 or y = 150, so the largest area occurs when x = 75 and y = 150 so
A = xy = 75*150 = 11250 sq ft. y = 300-2x is your function which can also be written as
A(x) = 300 - 2x

Math Prof