Answer
a) $A(x)=-2.5x^2+375x$
b) $14062.5$ $ft^2$
Work Step by Step
a) Let $x$ be the length of the side of the rectangle that is parallel to three inner rows. Therefore, the total length of those 2 sides and three rows is $5x$. Because the farmer has only $750$ ft of fence, the total length of the remaining two sides of the rectangle is $750-5x$. Since those two sides are equal, each side has a length of $\frac{750-5x}{2}$, or $375-2.5x$.
The total area of 4 pens, as we can see in the illustration, is simply the area of the big rectangle. Because for rectangles, $area= length * width$, the area of 4 pens is:
$A(x)=(375-2.5x)x$
$A(x)=-2.5x^2+375x$
(with $a=-2.5$, $b=375$, $c=0$)
b) Because the model is a quadratic function, we can use our minimum/maximum formula to solve the problem.
The maximum value of the function occurs at:
$x=\frac{-b}{2a}=\frac{-375}{-5}=75$
Therefore, the largest possible total area is:
$A(75)=-2.5*75^2+375*75=-14062.5+28125=14062.5$
