Question 738857
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You are in luck.  The given number of fence panels is divisble by 4 which means that it is possible to maximize the area without compromising so that the panels fit.


Let *[tex \LARGE P] represent the available perimeter, i.e. the total length of fencing that you have.  Let *[tex \LARGE l] represent the length of your rectangle, and let *[tex \LARGE w] represent the width of the rectangle.


Since perimeter, length, and width are related thusly:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ P\ =\ 2l\ +\ 2w]


we can define length in terms of width and perimeter thus:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ l\ =\ \frac{P}{2}\ -\ w]


Since Area is length times width, using the above expression for *[tex \LARGE l] we can write a function that yields area as a function of width.


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ A(w)\ =\ -w^2\ +\ \frac{P}{2}w]


The astute student should recognize this function as a quadratic in standard form with coefficients *[tex \LARGE a\ =\ -1], *[tex \LARGE b\ =\ \frac{P}{2}], and *[tex \LARGE c\ =\ 0].  You should also note that, given the negative lead coefficient, the parabolic graph opens downwards meaning that the value of the function at the vertex is a maximum.


Using the formula for the *[tex \LARGE w]-coordinate of the vertex:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ w_v\ =\ \frac{-b}{2a}\ =\ \frac{-\frac{P}{2}}{2(-1)}\ =\ \frac{P}{4}].


So the width, in terms of the available perimeter, that yields the greatest area is the perimeter divided by 4.  That means that 2 times the width is the perimeter divided by 2.  Subtracting the perimeter divided by 2 from the perimeter, that leaves the perimeter divided by 2 which represents 2 times the length.  Hence, the length of the maximum area rectangle is also *[tex \LARGE \frac{P}{4}].


Therefore, for a given perimeter, the maximum area rectangle that can be constructed is a square with sides that measure one fourth of the perimeter.


John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
<font face="Math1" size="+2">Egw to Beta kai to Sigma</font>
My calculator said it, I believe it, that settles it
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