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 $A^{\prime }=700-4w=0$ 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 $\boxed{{\displaystyle 61250}}$ square meters, with dimensions $\boxed{{\displaystyle 175}}$ meters by $\boxed{{\displaystyle 350}}$ meters.