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Suppose you have 52 feet of fencing to enclose a rectangular dog pen. The function A = 26x – x2, where x = width, gives you the area of the dog pen in square feet. What width gives you the maximum area? What is the maximum area? Round to the nearest tenth as necessary. width = 26 ft; area = 364 ft2 width = 26 ft; area = 169 ft2 width = 13 ft; area = 507 ft2 width = 13 ft; area = 169 ft2

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sqdancefan

Answer:

  width = 13 ft; area = 169 ft^2

Step-by-step explanation:

The equation for the area can be factored as ...

  A = x(26 -x)

This has zeros at x=0 and x=26. The vertex of this downward-opening parabola will be halfway between those, at x=13. Then the area for a width of 13 feet is ...

  A = 13(26 -13) = 13^2 = 169 . . . . ft^2

The maximum area is 169 ft^2, when the width is 13 ft.

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You will note that these dimensions make the pen be a square. It can be useful to remember that a square is the rectangle with the largest area for a given perimeter length.

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