create a piecewise function word problem using quadratic equation

Sure! Here's a word problem involving a piecewise function with a quadratic equation:

**Word Problem:**

A city park has a rectangular grassy field where people can play sports. The field is surrounded by a walking track. The park management wants to place a water fountain in the field so that it is easily accessible to the visitors.

The accessibility of the fountain is modeled by a piecewise function \( A(x) \) that depends on the distance \( x \) from the center of the field. 

- **Case 1:** If a visitor is within 10 meters of the fountain (i.e., \( 0 \leq x \leq 10 \)), the accessibility is modeled by a quadratic equation \( A(x) = -0.1x^2 + 5x + 50 \).
- **Case 2:** If a visitor is more than 10 meters away but less than or equal to 30 meters from the fountain (i.e., \( 10 < x \leq 30 \)), the accessibility decreases linearly according to \( A(x) = -2x + 100 \).

- **Case 3:** If a visitor is more than 30 meters away (i.e., \( x > 30 \)), the accessibility is almost negligible, modeled by the constant function \( A(x) = 0 \).

**Question:**
- Write the piecewise function that represents the accessibility \( A(x) \) based on the distance \( x \) from the fountain.
- Determine the accessibility at \( x = 8 \), \( x = 15 \), and \( x = 35 \).

**Solution:**

The piecewise function can be written as:

\[
A(x) =
\begin{cases} 
-0.1x^2 + 5x + 50 & \text{if } 0 \leq x \leq 10 \\
-2x + 100 & \text{if } 10 < x \leq 30 \\
0 & \text{if } x > 30 
\end{cases}
\]

Let's calculate the accessibility at different distances:

1. **At \( x = 8 \) meters:** 
   - Since \( 0 \leq 8 \leq 10 \), use the quadratic equation.
   - \( A(8) = -0.1(8)^2 + 5(8) + 50 = -0.1(64) + 40 + 50 = -6.4 + 40 + 50 = 83.6 \)
   - So, the accessibility at 8 meters is **83.6**.

2. **At \( x = 15 \) meters:** 
   - Since \( 10 < 15 \leq 30 \), use the linear equation.
   - \( A(15) = -2(15) + 100 = -30 + 100 = 70 \)
   - So, the accessibility at 15 meters is **70**.

3. **At \( x = 35 \) meters:**
   - Since \( x > 30 \), the accessibility is given by the constant function.
   - \( A(35) = 0 \)
   - So, the accessibility at 35 meters is **0**.

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Would you like further details or have any questions?

### Related Questions:
1. What is the maximum accessibility value, and at what distance does it occur?
2. For what distance \( x \) is the accessibility exactly 50?
3. How would the function change if the maximum distance for the linear decrease was 40 meters instead of 30 meters?
4. Can you determine the rate of change of accessibility within the first 10 meters?
5. What would the piecewise function look like if the quadratic equation had a higher leading coefficient?
6. If the fountain was moved, making the quadratic part range from 0 to 15 meters, how would the piecewise function change?
7. How does the accessibility change between 10 and 30 meters compared to the first 10 meters?
8. Determine where the quadratic function has its vertex within the 0 to 10 meters range.

### Tip:
In piecewise functions, it's important to pay attention to the domain of each piece to correctly apply the function to different input values.

Explore how to create a piecewise function word problem using quadratic equations. Learn about piecewise functions and quadratic equations with a detailed example. Suitable for high school students studying algebra and functions.

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