Final answer:
The dimensions that will maximize the area of the farm, given 200 feet of fencing, are Length = 100 feet and Width = 50 feet.
Step-by-step explanation:
To maximize the area of a rectangular farm using a fixed amount of fencing materials, we can use calculus or understand that the maximum area of a rectangle with a given perimeter is a square. In this case, however, since one side of the rectangle is along the river and does not require fencing, we should look for a scenario where the remaining three sides use the available fencing optimally. We have to allocate the 200 feet of fencing to two widths and one length of the rectangle.
The perimeter of the rectangle that requires fencing is represented by 2w + l = 200, where w is the width and l is the length. To find the dimensions that give the maximum area A, we express the length l in terms of the width w: l = 200 - 2w. Then, the area of the rectangle is A = w * l = w * (200 - 2w).
To find the maximum area, we can set the derivative of A with respect to w to zero: A'(w) = 200 - 4w. Setting A'(w) to zero gives us w = 50. By substituting back into the formula for l, we get l = 200 - 2*50 = 100. Therefore, the dimensions that will maximize the area are Length = 100 feet, and Width = 50 feet.
Comparing the given options, the correct choice would be A: Length = 100 feet, Width = 50 feet since this combination uses the entire 200 feet of fencing and provides the largest possible area for the farm.