# A farmer owns 1000 meters of fence, and wants to enclose the largest possible rectangular area. The region to be fenced has a straight canal on one side, and thus needs to be fenced on only three sides. What is the largest area she can enclose?

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To enclose the largest possible rectangular area with 1000 meters of fence and a canal on one side, the farmer should use the entire fence length for the three sides perpendicular to the canal. This creates a rectangle with two equal sides and one shorter side.

Let's call the length of the two equal sides x and the length of the shorter side y. Since the total length of the fence is 1000 meters, the perimeter of the rectangle is (2x + y = 1000).

To maximize the area, we need to maximize the product of x and y. We can express y in terms of x as (y = 1000 - 2x), and then substitute this expression for y into the area formula (A = xy).

So, (A = x(1000 - 2x)).

To find the maximum area, we need to find the critical points of this function, which is the derivative of A with respect to x.

(A' = 1000 - 4x).

Setting (A') equal to zero and solving for x:

(1000 - 4x = 0),

(4x = 1000),

(x = 250).

So, one side of the rectangle is 250 meters.

Substitute x = 250 into the expression for y:

(y = 1000 - 2(250) = 500).

Therefore, the largest possible area the farmer can enclose is (250 \times 500 = 125000) square meters.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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