A farmer has 600 yards of fence. He will use some of the fence to enclose a rectangular area. He will use the rest divide the area into two congruent rectangles. What is the value of x that results in largest area? What is the largest area that the farmer can enclose? What are the length and the width of the outer fence that will produce the largest total area?

A pig pen is bounded on one side by a barn. The pen is to be enclosed on three sides by a fence, to create a retangular enclosure. The total length of the fence is to be used is 40m. What dimensions are needed to enclose an rea of 198m^3 in the pen?

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To find the value of x that results in the largest area, we can use the concept of derivatives. Let's break down the problem step by step:

1. Let's assume the length of the outer rectangle is L and the width is W. We also need to divide the area into two congruent rectangles, so the width of each of the congruent rectangles will be W/2.

2. We know that the perimeter of the outer rectangle is 600 yards. Since the perimeter of a rectangle is given by the formula: P = 2L + 2W, we can substitute the values:

600 = 2L + 2W

Simplifying further: 300 = L + W

3. We need to express the area of the outer rectangle in terms of the variables L and W. The area of a rectangle is given by the formula: A = L * W. Substituting L = 300 - W, we get:

A = (300 - W) * W = 300W - W^2

4. To find the value of x that results in the largest area, we can take the derivative of the area equation with respect to W and set it to zero. Differentiating A = 300W - W^2 with respect to W gives us:

dA/dW = 300 - 2W = 0

Solving for W, we get W = 150.

5. With W = 150, we can substitute it back into the equation L = 300 - W. So L = 300 - 150 = 150.

Now we have the length L = 150 and the width W = 150, which will maximize the area.

6. Finally, we can calculate the largest area by substituting these values into the area equation:

A = L * W = 150 * 150 = 22500 square yards.

Therefore, the value of x (or W) that results in the largest area is 150, the largest area that the farmer can enclose is 22500 square yards, and the length and width of the outer fence that will produce the largest total area are both 150 yards.