Answer
Model: $A(x)=-2x^2+2400x$
Rear side: $600$ ft; Top side: $1200$ ft
Work Step by Step
First, we need to build a quadratic function $f(x)$. Let $x$ be the length of the sides that are perpendicular to the river. Because there are two sides like that, their total length is $2x$. The question informs us that the farmer only has $2400$ ft of fence, and he only needs to build one more side that is opposite to the river. Therefore, the length of the remaining side is $2400-2x$.
The area of the rectangle is calculated by multiplying the length of two adjacent sides. In this case, they are $x$ and $2400-2x$. Hence, we can build an equation, with $A(x)$ being the area of the fenced region:
$A(x)=x(2400-2x)$
$A(x)=-2x^2+2400x$
(with $a=-2$, $b=2400$, $c=0$)
Using the model, we can utilize our formula regarding the maximum/minimum value of a quadratic function to answer the question.
The maximum value occurs at:
$x=\frac{-b}{2a}=\frac{-2400}{-4}=600$
Therefore, the rear sides of the area have a length of 600 ft. The top side has the length of:
$2400-2*600=2400-1200=1200$ ft
