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### Problem Breakdown:
Mario is building a rectangular pen where:
- One side of the pen is against the barn (so no fencing is needed there).
- The other three sides will use wire fencing.
- Mario has 900 feet of fencing available.

We need to:
1. **Find an equation for the area of the pen in terms of \( w \)**, the length perpendicular to the barn.
2. **Maximize the area** given that Mario has 900 feet of fencing.

### Part a) Writing the Area Equation:
Let:
- \( w \) be the width of the pen (the side perpendicular to the barn).
- \( l \) be the length of the pen (parallel to the barn).

For fencing:
- The total fencing will be used for the two widths (2 sides) and one length (1 side), so:
  \[
  2w + l = 900 \quad \text{(since we only fence three sides)}
  \]
  Solving for \( l \):
  \[
  l = 900 - 2w
  \]

The **area** of the pen is the product of length and width:
\[
A = l \times w = w \times (900 - 2w)
\]
Thus, the area function in terms of \( w \) is:
\[
A(w) = w(900 - 2w) = 900w - 2w^2
\]
This is a quadratic equation for the area.

### Part b) Maximizing the Area:
To maximize the area, we find the critical points of \( A(w) \). Since the equation is quadratic, the maximum occurs at the vertex. The formula for the vertex \( w \) of a quadratic function \( A(w) = -2w^2 + 900w \) is:
\[
w = \frac{-b}{2a} = \frac{-900}{2(-2)} = 225
\]
So, the width \( w \) that maximizes the area is \( 225 \) feet.

Finally, the corresponding length is:
\[
l = 900 - 2(225) = 450 \, \text{feet}
\]

Thus, the dimensions that maximize the area are:
- Width \( w = 225 \) feet,
- Length \( l = 450 \) feet.

---

Would you like further clarification on any step? Here are five related questions that might help expand your understanding:

1. What is the general formula for the vertex of a quadratic equation?
2. How would the solution change if Mario had more or less than 900 feet of fencing?
3. Can you explain why the maximum occurs at the vertex of the quadratic function?
4. What if all four sides needed fencing—how would that change the area function?
5. How do you confirm if a quadratic function is maximized or minimized at its vertex?

**Tip:** In optimization problems, quadratic equations often arise, and finding the vertex is key to solving them efficiently!

Mario wants to build a rectangular pen for his animals. One side of the pen will be against the barn; the other three sides will be enclosed with wire fencing. If Mario has 900 feet of fencing, what dimensions would maximize the area of the pen?

This problem involves optimizing the dimensions of a rectangular pen given a fixed amount of fencing. Using quadratic equations, learn how to maximize the area when one side of the pen is against the barn and only three sides are fenced. This solution is suitable for high school students in Grades 9-12.

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