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Wyoli plans to enclose a rectangular area for his sheep to graze, with approximately 200 feet of fencing. One side will be bordered by a straight section of river, the other three by the fence, and the enclosure has a width of
xft, as seen in the figure. Which of the following functions best models the enclosed area, in square feet?

Wyoli plans to enclose a rectangular area for his sheep to graze, with approximately $200$ feet of fencing. One side will be bordered by a straight section of river, the other three by the fence, and the enclosure has a width of $x$ ft, as seen in the figure. Which of the following functions best models the enclosed area, in square feet?

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Evaluate two-variable equations: word problems

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Q. Wyoli plans to enclose a rectangular area for his sheep to graze, with approximately

200

feet of fencing. One side will be bordered by a straight section of river, the other three by the fence, and the enclosure has a width of

x

ft, as seen in the figure. Which of the following functions best models the enclosed area, in square feet?

Understand the problem: Understand the problem. $\newline$ We need to find a function that models the area of a rectangular enclosure where one side is a river and the other three sides are made of fencing. The total length of the fencing is $200$ feet, and the width of the enclosure is $x$ feet.
Set up equation for perimeter: Set up the equation for the perimeter. $\newline$ Since one side of the rectangle is bordered by the river, we only need to consider the fencing for the other three sides. The perimeter made by the fencing is $200$ feet. If we let $L$ be the length of the side opposite the river, the equation for the perimeter is: $\newline$ $L + 2x = 200$
Solve for L: Solve for L in terms of $x$ . $L = 200 - 2x$
Write area function: Write the function for the area of the rectangle. $\newline$ The area $A$ of a rectangle is given by the formula $A = \text{length} \times \text{width}$ . We have the width as $x$ and the length as $(200 - 2x)$ . So the function for the area is: $\newline$ $A(x) = x \times (200 - 2x)$
Simplify area function: Simplify the function for the area. $\newline$ $A(x) = 200x - 2x^2$ $\newline$ This function models the area of the enclosure in terms of the width $x$ .

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