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 city is organizing a two-stage annual firework show on New Year's Eve. The city plans to launch fireworks from two different locations, and the height of the fireworks is modeled by two different piecewise functions depending on the time of the launch. Let \( y \) represent the height (in feet) of the fireworks above the ground, and \( x \) represent the time (in seconds) after the first firework is launched.

1. **First Stage:** The first set of fireworks is launched from Location A. The height of these fireworks follows the equation:

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

   This equation models the height of the fireworks launched at Location A between 450 and 1550 seconds after the first firework is launched.

2. **Second Stage:** After a brief pause, the second set of fireworks is launched from Location B. The height of these fireworks follows a different equation:

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

   This equation models the height of the fireworks launched at Location B between 2450 and 3550 seconds after the first firework is launched.

**Questions:**
1. What is the maximum height reached by the fireworks at Location A?
2. What is the time at which the maximum height is reached for Location A?
3. What is the maximum height reached by the fireworks at Location B?
4. What is the time at which the maximum height is reached for Location B?
5. For how many seconds are the fireworks visible in the first stage?
6. For how many seconds are the fireworks visible in the second stage?
7. What is the height of the fireworks at \( x = 450 \) seconds?
8. What is the height of the fireworks at \( x = 3550 \) seconds?

**Tip:** Remember that the vertex of a parabola \( y = a(x - h)^2 + k \) gives the maximum or minimum point of the function, which occurs at \( x = h \).

Explore a word problem involving piecewise functions and quadratic equations. Learn how to model the height of fireworks launched from two different locations over time, including finding maximum heights, times of maximum heights, and heights at specific times. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation