To proceed with the calculations for the roller coaster track design, let's break down the requirements in Part A of your task:

1. Define the Linear Equations for L1 and L2:

   • You need to create two linear equations for the ascent (L1) and drop (L2) segments of the track.
   • The slope for L1 (ascent) is given as $0.7$, and the slope for L2 (drop) is $-1.8$.

2. Establish a Parabolic Equation for Smooth Transition:

   • You will connect the linear segments L1 and L2 with a parabolic segment $f(x)$.
   • Ensure a smooth transition by setting the tangents of the parabola at points $P$ and $Q$ equal to the slopes of L1 and L2.
   • The horizontal distance between $P$ and $Q$ is set at 16 meters, with the origin placed at $P$.

Steps for Part A Calculations

1. Setting Up Equations for L1 and L2

For a line $y = mx + c$:

• For L1, the slope $m = 0.7$, so $y_1(x) = 0.7x + c_1$.
• For L2, the slope $m = -1.8$, so $y_2(x) = -1.8x + c_2$.

Since the origin is at $P$ (starting point for L1), we can take $c_1 = 0$ to simplify, so: $y_1(x) = 0.7x$ For $y_2(x)$, we’ll need further calculations based on the parabolic section.

2. Defining the Parabola $f(x)$

Assume $f(x)$ is of the form $f(x) = ax^2 + bx + c$.

• At $P (x=0)$, $f(x)$ meets $y_1(x) = 0.7x$, which implies $f'(0) = 0.7$.
• At $Q (x=16)$, $f(x)$ meets $y_2(x) = -1.8x + c_2$, so $f'(16) = -1.8$.

Using these conditions, we can solve for coefficients $a$, $b$, and $c$.

Once these equations are established and solved, you can extend to Part B by integrating additional segments to join L1 from the left and extend L2 to the right, following similar principles for continuity and smooth transitions.

Would you like me to continue with the detailed steps for the parabolic equation calculations?

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Follow-Up Questions:

1. How would you determine specific coefficients $a$, $b$, and $c$ for the parabolic section $f(x)$?
2. Are there constraints on the maximum or minimum height of the track you’d like to consider?
3. What methods are you considering for additional track sections in Part B?
4. Should the overall track length be optimized for any specific properties, like smoothness or safety?
5. Would you like help setting up a software plot to visualize the segments?

Tip: For roller coaster designs, consider symmetry or specific patterns in function shapes for visually appealing transitions.