Let's solve this problem step-by-step.

Problem Analysis

Farmer Ed wants to enclose a rectangular plot that borders a river, using only three sides of fencing (two widths and one length). He has 5,500 meters of fencing to use, and the side along the river will not be fenced.

Let:

• $x$ = the length of the two sides perpendicular to the river.
• The length along the river is $5500 - 2x$.

Step 1: Define the Area Function

The area $A$ of the rectangular plot can be expressed in terms of $x$: $A = x \times (5500 - 2x)$ Simplify the equation: $A = 5500x - 2x^2$

Step 2: Find the Maximum Area

To find the maximum area, we need to express $A$ as a quadratic function and determine its maximum value.

The function is: $A(x) = -2x^2 + 5500x$

Step 3: Find the Vertex of the Parabola

The quadratic function $A(x) = -2x^2 + 5500x$ is a downward-opening parabola. The maximum value occurs at the vertex. The formula for the vertex of a parabola $ax^2 + bx + c$ is: $x = -\frac{b}{2a}$ In our case:

• $a = -2$
• $b = 5500$

So, $x = -\frac{5500}{2 \times (-2)} = \frac{5500}{4} = 1375$

Step 4: Calculate the Maximum Area

Now substitute $x = 1375$ back into the area equation to find the maximum area: $A = 5500x - 2x^2$ $A = 5500(1375) - 2(1375)^2$ $A = 7562500 - 2 \times 1890625$ $A = 7562500 - 3781250 = 3781250$

Final Answer

The largest area that can be enclosed is $\boxed{3,781,250}$ square meters.

Would you like me to provide more details or solve any other parts of this problem?

Here are 5 related questions you might find interesting:

1. How would the solution change if Farmer Ed fenced all four sides instead of just three?
2. What if Farmer Ed had only 4,000 meters of fencing? How would that change the maximum area?
3. How does the concept of optimization apply to other practical problems in agriculture?
4. Can we use calculus to verify the maximum area we found for this problem?
5. What is the effect of changing the length of fencing available on the shape of the optimal rectangle?

Tip: When maximizing or minimizing a function, look for opportunities to use the vertex formula or calculus-based techniques like finding critical points.