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Section 1.11 Building a Roller Coaster
Subsection 1.11.1 Recommended Tutorials
Before starting on these exercises, you should familiarize yourself with the material covered in the following tutorials:
Subsection 1.11.2 Introduction
You are in charge of designing the first hill of a new roller coaster. For an initial design, you connect a straight stretch of track for the lift hill followed by two parabolas, as shown in
Figure 1.2 .
Figure 1.2. A simple design for the initial hill of a roller coaster.
The following criteria must be met to build the roller coaster:
The lift hill will have a slope of \(0.8\text{.}\)
The straight section of the lift hill will cover a horizontal distance of \(100\) ft.
The slope of the first descent will reach a maximum magnitude of \(1.6\) after another \(100\) ft.
The next hill will reach a height of \(50\) ft after another \(100\) ft.
The track must be smooth (i.e. there cannot be any sudden changes in the slope of the track).
The goal of the following exercises is to develop a piecewise function that satisfies all of the above criteria.
Exercises 1.11.3 Exercises
1. Let
\(L(x)\) be the function for the lift hill, which is a linear function passing through the origin. In this exercise, you will define
\(L(x)\) to satisfy the necessary roller coaster criteria.
(a) Assign the function
\(L(x)\) in Maple for the lift hill, using the given slope in
Figure 1.2 and a
\(y\) -intercept of zero.
(b) Evaluate
\(L(100)\) to determine the height at which this linear segment connects to the first parabolic segment.
2. Let
\(f(x)\) be the function for the first parabolic segment, opening downward. In this exercise, you will define
\(f(x)\) to satisfy the necessary roller coaster criteria.
(a) Assign the function
\(f(x)=ax^2+bx+c\text{,}\) since
\(f(x)\) must be a quadratic function.
Hint .
Be sure to include multiplication between adjacent variables.
(b)
In order to satisfy the necessary roller coaster criteria, you will need to solve a system of three equations:
\(f(100)\) must equal \(L(100)\) for the two segments to connect.
\(f'(100)\) must equal \(L'(100)\) for the track to be smooth (differentiable) at the connection point.
\(f'(200)=-1.6\text{,}\) according to the third criteria.
Using a single solve() or fsolve() command, solve this system of equations for \(a\text{,}\) \(b\text{,}\) and \(c\text{.}\)
(c) Reassign the function
\(f(x)\text{,}\) this time using the values of
\(a\text{,}\) \(b\text{,}\) and
\(c\) that you have just calculated.
3. Let
\(g(x)\) be the function for the second parabolic segment, opening upward. In this exercise, you will define
\(g(x)\) to satisfy the necessary roller coaster criteria.
(a) Assign the function
\(g(x)=px^2+qx+r\text{,}\) since
\(g(x)\) must be a quadratic function and you should avoid using the same coefficients that you used for
\(f(x)\text{.}\)
Hint .
Be sure to include multiplication between adjacent variables.
(b)
In order to satisfy the necessary roller coaster criteria, you will need to solve a system of three equations:
\(g(200)\) must equal \(f(200)\) for the two parabolic segments to connect.
\(g'(200)\) must equal \(f'(200)\) for the track to be smooth (differentiable) at the connection point.
\(g(300)=50\text{,}\) according to the fourth criteria.
Using a single solve() or fsolve() command, solve this system of equations for \(p\text{,}\) \(q\text{,}\) and \(r\text{.}\)
(c) Reassign the function
\(g(x)\text{,}\) this time using the values of
\(p\text{,}\) \(q\text{,}\) and
\(r\) that you have just calculated.
4.
Now that you have determined the functions of the three segments, it’s time for you to put them together as a piecewise function and plot your roller coaster. The piecewise function for the roller coaster on the interval \([0,300]\) is given by
\begin{equation*}
coaster(x) = \begin{cases}
L(x) \amp \text{ if } 0 \leq x \lt 100 \\
f(x) \amp \text{ if } 100 \leq x \lt 200 \\
g(x) \amp \text{ if } 200 \leq x \leq 300
\end{cases}\text{.}
\end{equation*}
(a) Use the
piecewise() command to define a piecewise function called
\(coaster(x)\text{.}\)
Hint .
An example of a piecewise function may be found in
Section 9.5 .
(b) Plot
\(coaster(x)\) over the interval
\([0,300]\text{.}\)
