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Part 1
A farmer has 1788 feet of fencing available to enclose a rectangular area bordering a river. If no fencing is required along the​ river, find the dimensions of the fenced area that will maximize the area. What is the maximum​ area?
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Part 1
What is the length of the fenced​ area?

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square feet
cubic feet
feet

To maximize the area of the rectangular enclosure with one side along the river (meaning only three sides require fencing), we can approach this problem by setting up the area function and using calculus to find the maximum.

Let:
- \( x \) be the length parallel to the river (the side that does not require fencing).
- \( y \) be the width perpendicular to the river (both sides of \( y \) require fencing).

### Step 1: Set up the perimeter equation
Since only three sides need fencing, the total fencing used is:
\[
x + 2y = 1788
\]

Rearrange for \( x \):
\[
x = 1788 - 2y
\]

### Step 2: Set up the area function
The area \( A \) of the rectangle is given by:
\[
A = x \cdot y
\]

Substitute \( x \) from the perimeter equation:
\[
A = (1788 - 2y) \cdot y
\]
\[
A = 1788y - 2y^2
\]

### Step 3: Differentiate and find the maximum
To maximize \( A \), take the derivative of \( A \) with respect to \( y \) and set it to zero:
\[
\frac{dA}{dy} = 1788 - 4y = 0
\]

Solve for \( y \):
\[
4y = 1788
\]
\[
y = 447
\]

### Step 4: Find \( x \)
Substitute \( y = 447 \) back into the equation for \( x \):
\[
x = 1788 - 2(447)
\]
\[
x = 1788 - 894
\]
\[
x = 894
\]

### Step 5: Calculate the maximum area
\[
A = x \cdot y = 894 \cdot 447 = 399,318 \text{ square feet}
\]

### Answer:
The dimensions of the fenced area that maximize the area are:
- Length (along the river): \( x = 894 \) feet
- Width (perpendicular to the river): \( y = 447 \) feet

The maximum area is \( 399,318 \) square feet.

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#### Related Questions
1. How would the solution change if all four sides required fencing?
2. What if the river was not straight, altering the shape of the fenced area?
3. How can we verify that this solution maximizes the area?
4. What if there were only 1000 feet of fencing available?
5. How does this method relate to real-world applications in optimizing resources?

#### Tip
In optimization problems, always express one variable in terms of the other to simplify calculations.

A farmer has 1788 feet of fencing available to enclose a rectangular area bordering a river. If no fencing is required along the river, find the dimensions of the fenced area that will maximize the area. What is the maximum area?

Solve an optimization problem involving a farmer with 1788 feet of fencing along a river. Learn how to set up the perimeter and area equations, differentiate, and find maximum dimensions for optimal area. Ideal for calculus students in Grades 10-12.

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