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Diana has available 280 yards of fencing and wishes to enclose a rectangular area.
(a) Express the area A of the rectangle as a function of the width W of the rectangle.
(b) For what value of W is the area largest?
(c) What is the maximum area?

1 Answer

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Answer:

(a) A = W(140 -W)

(b) W = 70

(c) 4900 square yards

Explanation:

You want an expression for the rectangular area that can be enclosed by 280 yards of fence, if that area has width W. Further, you want the value of W that maximizes the area, and the size of that maximum area.

Perimeter expression

The perimeter of the rectangle with length L and width W is ...

P = 2(L +W)

Then the length can be found from the width as ...

L = P/2 -W

For P=280, this is ...

L = 140 -W

(a) Area expression

The area is the product of length and width:

A = LW

A = (140 -W)(W)

(b) Maximum area

The area expression describes a downward-opening parabola with zeros at W=0 and W=140. The vertex (maximum) of the parabola is on the line of symmetry, halfway between these zeros. The value of W there is ...

W = (0 +140)/2 = 70

The area is largest for W = 70.

(c) Area

For that value of W, the area is ...

A = (140 -70)(70) = 70² = 4900 . . . . . square yards

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Additional comment

The generic solution to the area problem when fence is on all sides is that the shape is a square.

If the cost of fence is different for the different sides, the total cost in the east-west direction is equal to the total cost in the north-south direction when cost is minimized.

User Kalu Singh Rao
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