Make a word problem piecewise function using
y=−0.001 (x−1000)² + 3000 {450≤x≤1550}
y=−0.001(x−3000)² +3000 {2450≤x≤3550}

**Word Problem:**

A large amusement park hosts two major roller coaster rides that attract thrill-seekers from all around. The height of the roller coaster cars during the main drops on these two rides is modeled by a piecewise function. Let \( y \) represent the height (in feet) of the roller coaster car above the ground, and \( x \) represent the time (in seconds) since the car reached the top of the first drop.

1. **First Roller Coaster - "The Falcon's Plunge":** 

   The height of the roller coaster car during the main drop on "The Falcon's Plunge" is modeled by the equation:

   \[
   y = -0.001(x - 1000)^2 + 3000 \quad \text{for} \quad 450 \leq x \leq 1550
   \]

   This equation describes how the height of the roller coaster car changes over time during the main drop, which occurs between 450 and 1550 seconds after the ride begins.

2. **Second Roller Coaster - "The Eagle's Dive":**

   The height of the roller coaster car during the main drop on "The Eagle's Dive" is modeled by a different equation:

   \[
   y = -0.001(x - 3000)^2 + 3000 \quad \text{for} \quad 2450 \leq x \leq 3550
   \]

   This equation describes the height of the roller coaster car during the main drop, which occurs between 2450 and 3550 seconds after the ride begins.

**Questions:**
1. What is the highest point reached by the roller coaster car on "The Falcon's Plunge" during its main drop?
2. At what time does the roller coaster car on "The Falcon's Plunge" reach its highest point?
3. What is the highest point reached by the roller coaster car on "The Eagle's Dive" during its main drop?
4. At what time does the roller coaster car on "The Eagle's Dive" reach its highest point?
5. How long does it take for the car on "The Falcon's Plunge" to complete its main drop?
6. How long does it take for the car on "The Eagle's Dive" to complete its main drop?
7. What is the height of the roller coaster car at the start of "The Falcon's Plunge"?
8. What is the height of the roller coaster car at the start of "The Eagle's Dive"?

**Tip:** Notice that both roller coasters reach their highest point when \( x = 1000 \) for "The Falcon's Plunge" and \( x = 3000 \) for "The Eagle's Dive," as these values represent the vertices of the parabolas.

Explore a word problem involving piecewise functions and quadratic equations describing roller coaster heights. Solve for highest points, times, and durations of two roller coasters' main drops. Suitable for Grades 10-12.

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