an teacher wants to use 19 feet of scalloped border to block

an teacher wants to use 19 feet of scalloped border to block off a rectangular region on her bulletiin board. What is the max area that she can enclose with the border?

Solution

The perimeter must be 19 feet, so:
P=2*length + 2*width = 19
length + width = 9.5
length = 9.5-width
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Area = (length)(width)
Area = (9.5-w)(w)
Area = 9.5w-w^2
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This is a quadratic with a=-1,b=9.5
The maximum point is at w=-b/2a= 9.5/2= 4.75
So width is 4.75 ft.
length = 9.5-4.75 = 4.75 ft.
The teacher will get the maximum area if she lets the rectangle
be a square with a side of 4.75 ft.