We have a fence as described in the picture.
The enclosed area is equal to x*y, while the fence perimeter is 2x+y.
As we know that the fence is 500 ft long, we can write:
[tex]2x+y=500[/tex]
We have to maximize the area, subject to the constraint of the fence length of 500 ft.
We can use the constraint to replace y in the area equation:
[tex]\begin{gathered} 2x+y=500 \\ y=500-2x \end{gathered}[/tex][tex]xy=x(500-2x)=500x-2x^2[/tex]
Then, we have this objective function we need to maximize:
[tex]A(x)=500x-2x^2[/tex]
To maximize it we can derive A relative to x and equal it to 0 in order to find the value of x that maximizes A:
[tex]\begin{gathered} \frac{dA}{dx}=0 \\ 500(1)-2(2x)=0 \\ 500-4x=0 \\ 500=4x \\ x=\frac{500}{4} \\ x=125 \end{gathered}[/tex]
The value of y can be calculated as:
[tex]\begin{gathered} y=500-2x \\ y=500-2(125) \\ y=500-250 \\ y=250 \end{gathered}[/tex]
Answer:
b) The objective function is the enclosed area, and the constraint is the fence length F = 500 ft.
c) I choose to write the equation in terms of x and the objective function is 500x - 2x².
d) x = 125
y = 250
