A rectangular plot of farmland will be bounded on one side by a river and on the other three sides by a single-strand electric fence. With $700 \mathrm{~m}$ of wire at your disposal, what is the largest area you can enclose, and what are its dimensions?

Step 1 ：The problem is asking for the maximum area that can be enclosed with a given amount of fencing. This is a problem of optimization, which can be solved using calculus.

Step 2 ：The area of a rectangle is given by the formula \(A = l \times w\), where \(l\) is the length and \(w\) is the width. In this case, the length is the side along the river and the width is the side enclosed by the fence.

Step 3 ：Since the fence encloses three sides of the rectangle, the total length of the fence is \(2w + l = 700\). We can solve this equation for \(l\) to get \(l = 700 - 2w\).

Step 4 ：Substituting this into the area formula gives \(A = w(700 - 2w) = 700w - 2w^2\).

Step 5 ：To find the maximum area, we need to find the maximum of this function. This occurs where the derivative is zero, so we need to solve the equation \(A' = 700 - 4w = 0\) for \(w\).

Step 6 ：Solving the equation gives \(w = 175\) and \(l = 350\).

Step 7 ：Substituting these values into the area formula gives \(A = 175 \times 350 = 61250\).

Step 8 ：Final Answer: The maximum area that can be enclosed is \(\boxed{61250}\) square meters, with dimensions \(\boxed{175}\) meters by \(\boxed{350}\) meters.