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Answer:
A rectangular fence of width [tex]600[/tex] feet and length [tex]600[/tex] feet would achieve the maximum possible area: [tex]360\, 000[/tex] square feet.
Step-by-step explanation:
Let [tex]x[/tex] denote the width of the fence. These would account for two of the sides of the rectangle. There would be [tex](2400 - 2\, x)[/tex] of the fence remaining. Since these need to be evenly split between the other two sides of the rectangle, the "length" of the fence would need be [tex](1/2)\, (2400 - 2\, x) = 1200 - x[/tex].
The area of the rectangle would be the product of length and width:
[tex](x) (1200 - x) = -x^{2} + 1200\, x[/tex].
Note that since both width [tex]x[/tex] and length [tex](1200 - x)[/tex] need to be non-negative, [tex]0 < x < 1200[/tex].
When finding value of [tex]x[/tex] that would maximize or minimize the value of a continuous function, such as the area [tex](-x^{2} + 1200\, x)[/tex], within a given interval, the value of [tex]x[/tex] might be:
- Within the range of the interval, which requires the first derivative test, or
- If the interval is not an open interval: at the endpoints of the interval, which requires evaluating the function at these points.
Since in the given interval [tex]0 < x < 1200[/tex] , [tex]x[/tex] cannot be equal to either endpoint [tex]0[/tex] or [tex]1200[/tex], this interval would be an "open interval". It would not be necessary to evaluate the function at these two endpoints.
To find the values of [tex]x[/tex] that would maximize the area [tex](-x^{2} + 1200\, x)[/tex] within the range of the interval apply the first derivative test:
- Differentiate the expression for area with respect to [tex]x[/tex] to obtain the first derivative. Find values of [tex]x[/tex] that would set the first derivative to [tex]0[/tex].
- Evaluate the expression for area at each of these values of [tex]x[/tex]. Find the value of [tex]x[/tex] that achieves the maximum possible value.
The first derivative of area [tex](-x^{2} + 1200\, x)[/tex] with respect to width [tex]x[/tex] is:
[tex]\displaystyle \frac{d}{d x}\left[-x^{2} + 1200\, x\right] = -2\, x + 1200[/tex].
Solve for values of [tex]x[/tex] that would set this first derivative to [tex]0[/tex]::
[tex]-2\, x + 1200 = 0[/tex].
[tex]x = 600[/tex].
Evaluate the expression for area at this point:
[tex]-x^{2} + 1200\, x = -(600)^{2} + (1200)\, (600) = 360\, 000[/tex].
The length of the rectangular fence would be [tex](1200 - x) = 600[/tex].
In other words, a rectangular fence [tex]600[/tex] feet in width and [tex]600[/tex] feet in length would achieve the maximum possible area: [tex]360\, 000[/tex] square feet.
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