You have 200 feet of fencing to enclose a rectangular plot that borders on a river. If you do not fence the side along the river, find the length and width of the plot that will maximize the area. What is the largest area that can be enclosed?
The width, labeled X in the figure, is how many feet.

Step 1 ：Let the length of the plot be \(l\) and the width be \(w\). We have the equation \(l+2w=200\) because only three sides need to be fenced.

Step 2 ：Rearrange the equation to find \(l=200-2w\).

Step 3 ：We want to maximize the area of this rectangular plot, which is given by \(lw\). Substituting \(l\) into our expression for area, we have \[(200-2w)(w)=200w-2w^2\].

Step 4 ：We will now complete the square to find the maximum value of this expression. Factoring a \(-2\) out, we have \[-2(w^2-100w)\].

Step 5 ：In order for the expression inside the parenthesis to be a perfect square, we need to add and subtract \((100/2)^2=2500\) inside the parenthesis. Doing this, we get \[-2(w^2-100w+2500-2500) \Rightarrow -2(w-50)^2+5000\].

Step 6 ：Since the maximum value of \(-(w-50)^2\) is 0 (perfect squares are always nonnegative), the maximum value of the entire expression is 5000, which is achieved when \(w=50\) and \(l=200-2w=100\).

Step 7 ：Thus, the maximum area of the plot is \(\boxed{5000}\) square feet.