Question 733079
James has 500 feet of fencing to enclose a rectangular region on his farm for some sheep. A) Make a sketch of 3 possible regions that James could enclose and give the corresponding areas.

Comment:  The confusion comes from trying to relate part A to B,C,D etc.
I'm going to assume the "region" enclosed by the 500 ft of fencing.
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B) If the length of the region is x, find an expression for the width.
Perimeter = 2(Length + Width)
500 = 2(x + width)
250 = x + sidth
width = 250-x
 
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C) Use your answer from part B to write an equation for the area to the region.
Area = length*width = x(250-x) = 250x - x^2 sq. ft.

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D) Graph your equation from part C on your calculator(My teacher never told us to use one of those types of calculators) and sketch the graph.
{{{graph(400,400,-100,500,-100,17000,250x-x^2)}}}
 
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E) James wants his fenced region to have the largest area possible using 500 feet of fencing. Find this area using the graph or a table of values.
Use what ever method you know to find the "x" value that gives you
the maximum area where A(x) = x(250-x)
Ans: x = 125 ft. 
 
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F) What is the length and width of the region with the area from part E? Describe this region.
Ans: length = width = 125 ft.
The rectangle is a square. 
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Cheers,
Stan H.
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