You have 600 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?

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.

Step 2 ：The area of a rectangle is given by the formula \(A = lw\), where \(l\) is the length and \(w\) is the width. In this case, we only have to fence three sides of the rectangle, so the total length of the fencing is given by \(2w + l = 600\). We can solve this equation for \(l\) to get \(l = 600 - 2w\).

Step 3 ：We can then substitute this into the area formula to get \(A = w(600 - 2w)\). This is a quadratic function, and its maximum value occurs at the vertex. The x-coordinate of the vertex of a quadratic function given by \(f(x) = ax^2 + bx + c\) is \(-b/2a\). In this case, \(a = -2\) and \(b = 600\), so the maximum area occurs when \(w = -600/(2*-2) = 150\).

Step 4 ：We can then substitute \(w = 150\) into the equation \(l = 600 - 2w\) to find the corresponding length.

Step 5 ：Final Answer: The width of the plot that will maximize the area is \(\boxed{150}\) feet, the length is \(\boxed{300}\) feet, and the largest area that can be enclosed is \(\boxed{45000}\) square feet.