Make a word problem piecewise function using
y=−0.001 (x−1000)² + 3000 {450≤x≤1550}

Sure! Let's create a word problem involving a scenario where a piecewise function models a real-world situation. We'll use the function you've provided.

### Word Problem

**Scenario:**

A new shopping mall has opened with a unique architectural design. The height of the roof \( y \) above a walkway inside the mall varies along the length of the walkway \( x \). The walkway is designed in such a way that the roof's height forms a parabolic arc to maximize natural light and aesthetic appeal. 

**Section of the Walkway:**

The height \( y \) of the roof above the walkway can be described by the function:
\[ 
y = -0.001 (x - 1000)^2 + 3000 \quad \text{for} \quad 450 \leq x \leq 1550 
\]

This function models the roof's height \( y \) in centimeters, where \( x \) is the horizontal distance in centimeters from the start of the walkway.

### Questions

1. **What is the maximum height of the roof, and at what position along the walkway does it occur?**
   - *Hint:* Determine the vertex of the parabola described by the function.

2. **Calculate the height of the roof at the start (\(x = 450\)) and end (\(x = 1550\)) of the walkway.**
   - *Hint:* Plug the values of \(x\) into the function to find \(y\).

3. **How does the height of the roof change as you walk from the start to the end of the walkway?**
   - *Hint:* Consider the shape of the parabola and how it affects the height.

4. **At which position(s) is the roof's height exactly 2900 cm?**
   - *Hint:* Solve the equation \( y = 2900 \) for \(x\).

5. **Determine the roof's height at the midpoint of the walkway.**
   - *Hint:* Find the value of \(y\) at \( x = 1000 \).

6. **If the height of the roof must be at least 2800 cm to maintain architectural design standards, between which positions does this condition hold true?**
   - *Hint:* Solve the inequality \( y \geq 2800 \).

7. **Is the roof's height symmetric around its maximum point? Why or why not?**
   - *Hint:* Analyze the properties of the parabola and its vertex.

8. **How does the design of the roof help in maximizing natural light inside the mall?**
   - *Hint:* Consider the height and shape of the roof as well as its maximum point.

### Tip

When dealing with parabolic functions, the vertex form \( y = a(x-h)^2 + k \) provides useful information. The vertex \( (h, k) \) indicates the maximum or minimum point of the parabola. The coefficient \( a \) determines whether the parabola opens upwards or downwards and affects the width of the parabola.

Explore a word problem involving a piecewise function with a parabolic section. Understand how to determine maximum height, calculate heights at specific points, and analyze the shape of the parabola. Suitable for high school students.

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