The first thing we are going to do for this case is define variables.
We have then:
w: width
l: length
The perimeter is given by:
[tex] 2w + 2l = 520
[/tex]
The area is given by:
[tex] A = w * l
[/tex]
The area as a function of a variable is:
[tex] A (w) = w * (260-w)
[/tex]
Rewriting we have:
[tex] A (w) = -w ^ 2 + 260w
[/tex]
To obtain the maximum area, we derive:
[tex] A '(w) = -2w + 260
[/tex]
We equal zero and clear the value of w:
[tex] -2w + 260 = 0
2w = 260
[/tex]
[tex] w = \frac{260}{2}
w = 130
[/tex]
Then, the length is given by:
[tex] l = 260 - w
l = 260 - 130
l = 130
[/tex]
Finally, the maximum area obtained is:
[tex] A = w * l
A = 130 * 130
A = 16900
[/tex]
Answer:
A retangle that maximizes the enclosed area has a length of 130 yards and a width of 130 yards.
The maxium area is 16900 square yards
