Part A:
1st Linear track: L_1 (x)= 0.7x
Parabolic track: f(x)=-0.078125x^2+0.7x
2nd Linear track: L_2 (x)= -1.8x+ 20
Part B
The roller coaster design from the left of part A
Section 1 (Cubic Transition): f_(4 ) (x)=0.02x^2+0.7x
Section 2 (Cubic Transition):f_5 (x)= 0.011x^3+0.2x^2+0.7x
Section 3 (Flat section): L_6 (x)=0
The roller coaster (right of part A) design from x=10 to the exit is as follows:
Section 4 (Cubic Transition):
g(x)=−0.1875x3+7.1675x2−87.9625x+349.5625
Section 5 (Oscillatory Segment):
h(x)=5sin(0.4(x−15))+10
Section 6 (Final Descent):
j(x)=10e−0.2(x−20)
I like to see	Coefficients for the additional segments in Part B & plotting the full track

Here is the plotted design of the roller coaster track. The segments are as follows:

- **Part A** includes the initial linear track, parabolic curve, and another linear descent.
- **Part B** showcases a series of transitions, a flat section, an oscillatory segment, and a final descent.

Would you like additional details on any segment or the calculations for specific parts? 

### Related Questions:
1. What are the main differences between parabolic and cubic transition curves in roller coaster designs?
2. How do oscillatory functions like sine impact the safety of a roller coaster track?
3. Why might a designer choose an exponential descent for the final section of the ride?
4. What role do cubic transitions play in smoothing the coaster track between sections?
5. How does each type of function (linear, parabolic, cubic, oscillatory) affect the speed and experience of the ride?

**Tip:** When designing tracks, gradual changes like cubic transitions help to maintain ride smoothness.

Part A:
1st Linear track: L_1(x) = 0.7x
Parabolic track: f(x) = -0.078125x^2 + 0.7x
2nd Linear track: L_2(x) = -1.8x + 20

Part B:
The roller coaster design from the left of part A:
Section 1 (Cubic Transition): f_4(x) = 0.02x^2 + 0.7x
Section 2 (Cubic Transition): f_5(x) = 0.011x^3 + 0.2x^2 + 0.7x
Section 3 (Flat section): L_6(x) = 0

The roller coaster (right of part A) design from x = 10 to the exit:
Section 4 (Cubic Transition): g(x) = -0.1875x^3 + 7.1675x^2 - 87.9625x + 349.5625
Section 5 (Oscillatory Segment): h(x) = 5sin(0.4(x - 15)) + 10
Section 6 (Final Descent): j(x) = 10e^(-0.2(x - 20))

I would like to see coefficients for the additional segments in Part B & plotting the full track.

Explore a comprehensive roller coaster track design involving linear, parabolic, and cubic transitions, along with oscillatory and exponential descent segments. This detailed solution includes coefficients for each section, including cubic and oscillatory functions, and demonstrates full track plotting. Suitable for advanced high school mathematics students.

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