Answer
a) $A(x)=-x^2+1200x$
b) Length: $600$ ft, width: $600$ ft
Work Step by Step
a) The image has already given us the length in terms of $x$. Because the area of a rectangle is the product of width and length:
$A(x)=x(1200-x)$
$A(x)=-x^2+1200x$
(with $a=-1$, $b=1200$, $c=0$)
b) Using the minimum/maximum formula can help us solve this question.
The maximum value occurs at:
$x=\frac{-b}{2a}=\frac{-1200}{-2}=600$
Because $1200-x$ is our length, we replace $x$ with $600$ to find the length when the corral's area is maximized.
Our length is: $1200-600=600$ (ft)
Thus, in order to maximize the area, the fence's length must be $600$ ft, and the fence's width must also be $600$ ft.
