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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 200200 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 xx ft, as seen in the figure. Which of the following functions best models the enclosed area, in square feet?

Full solution

Q. Wyoli plans to enclose a rectangular area for his sheep to graze, with approximately 200200 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 xx ft, as seen in the figure. Which of the following functions best models the enclosed area, in square feet?
  1. Understand the problem: Understand the problem.\newlineWe 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 200200 feet, and the width of the enclosure is xx feet.
  2. Set up equation for perimeter: Set up the equation for the perimeter.\newlineSince 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 200200 feet. If we let LL be the length of the side opposite the river, the equation for the perimeter is:\newlineL+2x=200L + 2x = 200
  3. Solve for L: Solve for L in terms of xx.L=2002xL = 200 - 2x
  4. Write area function: Write the function for the area of the rectangle.\newlineThe area AA of a rectangle is given by the formula A=length×widthA = \text{length} \times \text{width}. We have the width as xx and the length as (2002x)(200 - 2x). So the function for the area is:\newlineA(x)=x×(2002x)A(x) = x \times (200 - 2x)
  5. Simplify area function: Simplify the function for the area. \newlineA(x)=200x2x2A(x) = 200x - 2x^2\newlineThis function models the area of the enclosure in terms of the width xx.

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