Question 438013
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You are over-thinking the setup here.


Imagine 3 rectangles stacked on top of one another creating another, larger rectangle.


Let the width of this larger rectangle be represented by *[tex \Large x].  Let the length of the larger rectangle (the sum of the widths of the three small ones) be represented by *[tex \Large y]


Now the perimeter of this larger rectangle is just *[tex \Large 2x\ +\ 2y] like any other rectangle, but remember that this is three rectangular areas so there are two more pieces of fence each of which measures *[tex \Large x], so we can write:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 2y\ +\ 4x\ =\ 340]


to show the relationship between the length and width of the large rectangle and the amount of available fencing.


Next, the area of the large rectangle, which is the area we are asked to maximize is given by:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ A(x,y)\ =\ xy]


But if we solve the fencing relationship for *[tex \Large y] we can create a function for the area in terms of the width:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ 170\ -\ 2x]


Then, by substitution:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ A(x)\ =\ -2x^2\ +\ 170x]


Which is a quadratic function of the form:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \rho(x)\ =\ ax^2\ +\ bx\ + c]


where


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ a\ =\ -2\ \ ], *[tex \LARGE b\ =\ 170\ \ ], and *[tex \LARGE c\ =\ 0]


Since this is a quadratic with a negative lead coefficient, the graph is a parabola that opens downward, hence the vertex represents the maximum value of the function.


Use the formula for the *[tex \Large x]-coordinate of the vertex of a parabola to find the overall width that maximizes the area:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x_v = \frac{-b}{2a}\ =\ \frac{-170}{2(-2)}\ =\ 42.5\ \ ]feet


Substitute this value into the equation for the fencing:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 2y\ +\ 4(42.5)\ =\ 340]


and solve for the overall length:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ 85\ \ ]feet


John
*[tex \LARGE e^{i\pi} + 1 = 0]
My calculator said it, I believe it, that settles it
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