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We are given a rectangular garden where one side is bordered by a river, so no fencing is needed on that side. The total length of the fencing available is 520 feet. Our goal is to determine the dimensions that will maximize the garden's area.

Let's define the following:
- Let \( x \) represent the length of the side perpendicular to the river (the shorter side).
- Let \( y \) represent the length of the side parallel to the river (the longer side).

### Step 1: Relationship between the perimeter and dimensions
Since no fencing is required along the river, the total length of fencing used will be for two shorter sides and one longer side:
\[
2x + y = 520
\]
Solve for \( y \):
\[
y = 520 - 2x
\]

### Step 2: Expression for the area
The area \( A \) of the rectangular garden is given by:
\[
A = x \times y
\]
Substitute \( y = 520 - 2x \) into the area formula:
\[
A = x(520 - 2x)
\]
\[
A = 520x - 2x^2
\]

### Step 3: Maximizing the area
To find the maximum area, we need to take the derivative of \( A \) with respect to \( x \) and set it equal to zero:
\[
\frac{dA}{dx} = 520 - 4x
\]
Set the derivative equal to zero to find the critical points:
\[
520 - 4x = 0
\]
\[
4x = 520
\]
\[
x = 130
\]

### Step 4: Calculate \( y \)
Now that we have \( x = 130 \), substitute it into the equation for \( y \):
\[
y = 520 - 2(130) = 520 - 260 = 260
\]

### Step 5: Calculate the maximum area
The maximum area is:
\[
A = 130 \times 260 = 33,800 \text{ square feet}
\]

### Final Answer:
- Shorter side: \( 130 \) feet
- Longer side: \( 260 \) feet
- Greatest possible area: \( 33,800 \) square feet

Would you like more details or have any further questions?

### Related Questions:
1. How does the amount of fencing affect the dimensions for maximum area?
2. What happens to the area if the garden was completely fenced (no river boundary)?
3. Can we generalize this method for any given length of fencing?
4. What if the garden needed fencing on all sides—how would the dimensions change?
5. How would the shape change if the river bordered two sides of the garden?

**Tip:** For maximizing or minimizing areas, calculus (derivatives) is a powerful tool to find optimal solutions.

A gardener has 520 feet of fencing to fence in a rectangular garden. One side of the garden is bordered by a river and does not need any fencing. What dimensions would guarantee that the garden has the greatest possible area?

Learn how to maximize the area of a rectangular garden bordered by a river using 520 feet of fencing. This problem covers key optimization concepts using algebra and quadratic functions, perfect for high school students. Includes a step-by-step guide to finding the garden's optimal dimensions.

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