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?
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.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.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: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.The area A of a rectangle is given by the formula A=length×width. We have the width as x and the length as (200−2x). So the function for the area is:A(x)=x×(200−2x)
Simplify area function: Simplify the function for the area. A(x)=200x−2x2This function models the area of the enclosure in terms of the width x.
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