SOLUTION: A rectangular pen is to be made with 120 feet of fencing. The pen is divided into 3 equal parts, and an existing property fence will be used for one of the long sides of the pen. (

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Question 844910: A rectangular pen is to be made with 120 feet of fencing. The pen is divided into 3 equal parts, and an existing property fence will be used for one of the long sides of the pen. (a) If x represents the width of the fence, express its area A(x) in terms of x. (b) Graph the function (c) Determine the dimensions of the rectangle that will make the area maximum.
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
x for width and y for length;
The quantities of fence material will be 3 of y and 2 of x. A picture would help to see this. The sum of these quantities must be equal to the given amount of fence material, the 120 feet: 2x%2B3y=120.

The area A=xy.
Using the fence quantity equation, solve for y, and then substitute into the Area equation:
3y=120-2x
y=40-%282%2F3%29x;
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A=x%2840-%282%2F3%29x%29
highlight_green%28A=40x-%282%2F3%29x%5E2%29, a parabola with a maximum.
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The zeros can be found using the previous, factored form. These zeros are at x=0 and at 40-%282%2F3%29x=0.
40=%282%2F3%29x
x=40%283%2F2%29
x=60
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The maximum must occur in the exact middle of x=0 and x=60, which is at highlight%28x=30%29.
Maximum Area is highlight%28highlight%2840%2A30-%282%2F3%29%2A30%5E2%29%29 square feet.