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This problem involves finding the dimensions of a rectangular garden that will maximize the area, given a fixed amount of fencing. The garden is bordered by a river on one side, so fencing is only needed for three sides.

### Step-by-Step Approach:

Let:
- \( x \) be the length of the side parallel to the river (no fence needed on this side).
- \( y \) be the length of the other two sides (perpendicular to the river), which need fencing.

The total length of the fence available is 400 feet, and fencing is required only on three sides. Therefore, the equation for the perimeter is:
\[
x + 2y = 400
\]
This simplifies to:
\[
x = 400 - 2y
\]

The area \( A \) of the garden is given by:
\[
A = x \times y
\]
Substitute \( x = 400 - 2y \) into the area equation:
\[
A = (400 - 2y) \times y = 400y - 2y^2
\]

### Maximize the Area:

To find the maximum area, take the derivative of \( A \) with respect to \( y \), and set it equal to zero:
\[
\frac{dA}{dy} = 400 - 4y
\]
Set \( \frac{dA}{dy} = 0 \):
\[
400 - 4y = 0
\]
Solving for \( y \):
\[
y = 100 \text{ feet}
\]

Now, substitute \( y = 100 \) into the equation for \( x \):
\[
x = 400 - 2(100) = 200 \text{ feet}
\]

### Conclusion:

- **Shorter side (perpendicular to the river)**: \( y = 100 \) feet
- **Longer side (parallel to the river)**: \( x = 200 \) feet
- **Greatest possible area**: 
\[
A = 200 \times 100 = 20,000 \text{ square feet}
\]

Would you like further details or have any questions? 

Here are 5 related questions for deeper understanding:
1. How would the solution change if all four sides of the garden needed fencing?
2. Can we derive a general formula for maximizing area with different amounts of available fencing?
3. How would the problem change if the river were on one of the shorter sides instead of the longer side?
4. What other mathematical methods can be used to solve this type of optimization problem?
5. What are the real-world applications of this type of problem?

**Tip**: Always check boundary conditions in optimization problems to ensure you find the maximum or minimum value.

A gardener has 400 feet of fencing to fence in a rectangular garden. One side of the garden is bordered by a river and so it does not need any fencing. What dimensions would guarantee that the garden has the greatest possible area?

Find the dimensions of a rectangular garden that maximize the area using 400 feet of fencing, with one side bordered by a river. Learn the optimization process through algebraic formulas and derivatives, suitable for Grades 9-12.

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