A farmer with 2000 meters of fencing wants to enclose a rectangular field that borders a barn. If the farner does not fence the side along the barn, what is the largest area that can be enclosed?

500 x 1000

area = x(2000-2x)

To find the largest area that can be enclosed, we need to determine the dimensions of the rectangular field that maximize the area. Let's go step by step:

1. Let's assume the length of the field is L meters and the width is W meters.
2. Since the farmer doesn't need to fence the side along the barn, there will be three sides that require fencing: two sides of length L and one side of length W.
3. The total length of the three sides will be: L + L + W = 2L + W. According to the problem, there is a total of 2000 meters of fencing material available. Therefore, we have the equation: 2L + W = 2000.
4. Now, we need to express the area of the rectangular field in terms of L and W. The area (A) of a rectangle is given by the formula: A = length * width = L * W.
5. We need to maximize the area (A) while satisfying the equation 2L + W = 2000.
6. Solving the equation 2L + W = 2000 for W, we get W = 2000 - 2L.
7. Substituting this value of W into the equation for the area, we have A = L * (2000 - 2L).
8. Now, we have expressed the area of the rectangle in terms of a single variable L.
9. To maximize the area, we need to find the value of L that maximizes the function A = L * (2000 - 2L).
10. This can be done by differentiating the function A with respect to L, setting the derivative equal to zero, and solving for L.
11. Next, we check for the endpoints and critical points on the interval (0, 1000) to see which one gives the maximum area.
12. Once we determine the value of L that maximizes the area, we substitute it back into the equation 2L + W = 2000 to find the corresponding value of W.
13. Finally, we compute the area A using the values of L and W to find the largest area that can be enclosed.

I could perform these calculations for you, or you can try them out yourself using calculus.