Answer:
The dimensions of the rectangular pen that maximize the area are [tex]w = 125\,ft[/tex] and [tex]l = 250\,ft[/tex].
Step-by-step explanation:
Let suppose that one side of the rectangular area to be fence coincides with the contour of the river, so that only three sides are needed to be enclosed. The equations of perimeter ([tex]p[/tex]) and area ([tex]A[/tex]), measured in feet and square feet, are introduced below:
[tex]p = 2\cdot w + l[/tex]
[tex]A = w\cdot l[/tex]
Where [tex]w[/tex] and [tex]l[/tex] are the length and width of the rectangle, measured in feet.
Besides, let suppose that perimeter is equal to the given amount of fencing, that is, [tex]p = 500\,ft[/tex]. The system of equations is:
[tex]2\cdot w + l = 500\,ft[/tex]
[tex]A = w\cdot l[/tex]
Let is clear the length of the rectangle and expand the area formula:
[tex]l = 500\,ft-2\cdot w[/tex]
[tex]A = w\cdot (500\,ft-2\cdot w)[/tex]
[tex]A = 500\cdot w -2\cdot w^{2}[/tex]
To determine the maximum area that can be enclosed, first and second derivatives to obtain the critical values that follow to an absolute maximum.
First derivative
[tex]A' = 500 - 4\cdot w[/tex]
Second derivative
[tex]A'' = -4[/tex]
Now, let equalize the first derivative to zero, the only critical value is:
[tex]500-4\cdot w = 0[/tex]
[tex]4\cdot w = 500[/tex]
[tex]w = 125\,ft[/tex]
Since the second derivative is a negative constant function, then, the previous outcome follows to an absolute maximum. The length of the rectangular area is: ([tex]w = 125\,ft[/tex])
[tex]l = 500\,ft - 2\cdot (125\,ft)[/tex]
[tex]l = 250\,ft[/tex]
The dimensions of the rectangular pen that maximize the area are [tex]w = 125\,ft[/tex] and [tex]l = 250\,ft[/tex].
