The perimeter for this case is given by:
[tex] P = y + 2x
[/tex]
Substituting values we have:
[tex] y + 2x = 950
[/tex]
The area is given by:
[tex] A = x * y
[/tex]
Writing the area based on a variable we have:
[tex] A (x) = x * (950 - 2x)
A (x) = -2x ^ 2 + 950x
[/tex]
We derive the area to obtain the maximum of the function:
[tex] A '(x) = -4x + 950
[/tex]
We equal zero and clear x:
[tex] -4x + 950 = 0
4x = 950
x = 950/4
x = 237.5
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Then, the other dimension is given by:
[tex] y = 950 - 2x
y = 950 - 2 * (237.5)
y = 475
[/tex]
Finally the maximum area is:
[tex] A = (237.5) * (475)
A = 112812.5 m ^ 2
[/tex]
Answer:
The length and width of the plot that will maximize the area are:
[tex] x = 237.5 m
y = 475 m
[/tex]
The largest area that can be enclosed is:
[tex] A = 112812.5 m ^ 2 [/tex]
