https://file.aiquickdraw.com/mathbot/file/1729940987594p62c8lo0.jpg

design a roller coaster and specify the equations for your roller coaster track from the starting point (entrance on the left) to its finish point (exit on the right). The three segments of track from Part A must be included in your track.
You must determine equations (minimum of 4) for the section on the left to join L1 and equations (minimum of 3) from  L2 to the exit point, ensuring a smooth transition between points-show detailed working, equations, constraints, reasons for the choices etc.
Part A: picture

To design a roller coaster track using the equations given in the image, we need to:

1. **Identify the existing segments** provided in Part A.
2. **Create additional equations** that join these segments smoothly.
3. **Ensure continuity** at each joint to maintain smooth transitions.

### Part A: Existing Track Segments

From the image, we observe three track segments:

1. **Linear Segment \( L_1 \):** \( y = 0.7x \)
2. **Quadratic Segment \( f(x) \):** \( f(x) = -0.078125x^2 + 0.7x \)
3. **Linear Segment \( L_2 \):** \( y = -1.8x + 20 \)

Each of these segments represents part of the roller coaster track.

### Plan for Additional Segments

To complete the track, we need to add:

1. **A section on the left of \( L_1 \)** that smoothly connects with \( L_1 \) (entrance).
2. **A section on the right of \( L_2 \)** that extends to the exit.

### 1. Design Equations for the Entrance Section

To create a smooth connection before \( L_1 \), we can introduce a new linear or quadratic function that connects to \( y = 0.7x \) with continuity in both value and slope at the transition point.

Assume the entrance is at \( x = -5 \). We’ll create two segments for the entrance:

#### Entrance Segment 1 (Parabolic)
Let's define this segment as:
\[ y = a_1x^2 + b_1x + c_1 \]

For smoothness, we need it to satisfy:
- \( y(-5) = y \) at the point where it meets \( L_1 \).
- \( y'(-5) = 0.

Design a roller coaster and specify the equations for your roller coaster track from the starting point (entrance on the left) to its finish point (exit on the right). The three segments of track from Part A must be included in your track. You must determine equations (minimum of 4) for the section on the left to join L1 and equations (minimum of 3) from L2 to the exit point, ensuring a smooth transition between points—show detailed working, equations, constraints, reasons for the choices, etc. Part A: picture.

Learn how to design a roller coaster track using linear and quadratic equations. This example includes smooth transitions and continuity between segments for a seamless track experience. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation