Fencing is limited to 20ft. in length. What is the maximum rectangular area that can be fenced inusing two perpendicular corner sides an existing wall

This problem involves maximizing the area of a rectangle where two sides are bordered by an existing wall, and the other two sides require fencing. The total length of fencing available is 20 feet, and you want to determine the maximum area that can be enclosed.

### Let's define:
- \( x \): the length of the side parallel to the wall.
- \( y \): the length of the side perpendicular to the wall.

Since the fence is used for only two sides (one parallel and one perpendicular to the wall), the total fencing used is:

\[
x + 2y = 20
\]

### Step 1: Express \( x \) in terms of \( y \)
Solving for \( x \) from the fencing constraint:

\[
x = 20 - 2y
\]

### Step 2: Express the area in terms of \( y \)
The area \( A \) of the rectangle is given by:

\[
A = x \cdot y
\]

Substitute \( x = 20 - 2y \) into this equation:

\[
A(y) = (20 - 2y) \cdot y
\]

\[
A(y) = 20y - 2y^2
\]

### Step 3: Maximize the area
To find the maximum area, we take the derivative of \( A(y) \) with respect to \( y \), and set it equal to zero:

\[
\frac{dA}{dy} = 20 - 4y = 0
\]

Solving for \( y \):

\[
4y = 20 \quad \Rightarrow \quad y = 5
\]

### Step 4: Find the corresponding \( x \)
Now, substitute \( y = 5 \) into the equation \( x = 20 - 2y \):

\[
x = 20 - 2(5) = 10
\]

### Step 5: Calculate the maximum area
The maximum area is:

\[
A = x \cdot y = 10 \cdot 5 = 50 \, \text{square feet}
\]

### Conclusion:
The maximum rectangular area that can be fenced using 20 feet of fencing along two perpendicular sides of an existing wall is **50 square feet**.

Do you have any questions or want further details?

Here are some related questions:
1. How would the solution change if you had more or less fencing?
2. What happens if the sides are not perpendicular to the wall?
3. Can this be solved using another optimization method?
4. What if you need to enclose the entire rectangle, with no wall used?
5. How does the shape of the rectangle affect the result?

**Tip**: When solving optimization problems, always check if you can use constraints to reduce the number of variables before differentiating.

Fencing is limited to 20ft. in length. What is the maximum rectangular area that can be fenced in using two perpendicular corner sides and an existing wall?

Learn how to maximize the area of a rectangle using 20ft of fencing along two sides and an existing wall. This problem involves optimization, geometry, and quadratic functions. Perfect for high school students, covering key concepts like the fencing constraint and maximizing area using derivatives.

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