DISCOVERY PROJECT

Designing a Roller Coaster

Suppose you are asked to design the first ascent and drop for a new roller coaster. By studying photographs of your favorite coasters, you decide to make the slope of the ascent $0.8$ and the slope of the drop $-1.6$. You then connect these two straight stretches $y = L_1(x)$ and $y = L_2(x)$ with part of a parabola $$y = f(x) = ax^2 + bx + c$$

where $x$ and $f(x)$ are measured in feet. For the track to be smooth there can't be abrupt changes in direction, so you want the linear segments $L_1$ and $L_2$ to be tangent to the parabola at the transition points $P$ and $Q$, as shown in the figure.

1. To simplify the equations, you decide to place the origin at $P$. As a consequence, what is the value of $c$?
2. Suppose the horizontal distance between $P$ and $Q$ is $100$ ft. To ensure that the track is smooth at the transition points, what should the values of $f'(0)$ and $f'(100)$ be?
3. If $f(x) = ax^2 + bx + c$, show that $f'(x) = 2ax + b$.
4. Use the results of problems 2 and 3 to determine the values of $a$ and $b$. That is, find a formula for $f(x)$.
5. Plot $L_1$, $f$, and $L_2$ to verify graphically that the transitions are smooth.



6. Find the difference in elevation between $P$ and $Q$.