To find the sinusoidal function that models the motion of the coaster, we need to express it in the form:

$y = A \sin(\omega t - \phi) + B$

Here, $y$ represents the height of the roller coaster above the ground, $A$ is the amplitude, $\omega$ is the angular frequency, $\phi$ is the phase shift, $B$ is the vertical shift, and $t$ is the time in seconds.

Step 1: Find the amplitude $A$

The amplitude $A$ is the distance from the middle of the sinusoidal curve to the maximum or minimum value. In this case, the maximum height is 106 feet (crest of the hill) and the minimum height is 6 feet (bottom of the hill).

The amplitude is half the distance between the maximum and minimum heights: $A = \frac{106 - 6}{2} = \frac{100}{2} = 50 \, \text{feet}$

Step 2: Find the vertical shift $B$

The vertical shift $B$ is the midpoint between the maximum and minimum heights: $B = \frac{106 + 6}{2} = \frac{112}{2} = 56 \, \text{feet}$

Step 3: Find the angular frequency $\omega$

It takes 1.8 seconds to go from the top of a hill to the bottom, which is half the period. The total period $T$ is the time it takes to go from one crest (top of the hill) to the next crest (or one full cycle). Therefore: $T = 2 \times 1.8 = 3.6 \, \text{seconds}$

The angular frequency $\omega$ is related to the period by the formula: $\omega = \frac{2\pi}{T} = \frac{2\pi}{3.6} \approx 1.745 \, \text{radians per second}$

Step 4: Find the phase shift $\phi$

Since the coaster starts at the top of the hill at $t = 0$, the sine function should have a phase shift to account for the fact that sine normally starts at 0. A cosine function would start at the maximum, but using sine, we need to account for the fact that we start at a crest, meaning a phase shift of $\frac{\pi}{2}$.

Thus, $\phi = -\frac{\pi}{2}$.

Step 5: Write the final equation

Now, we can plug in the values for $A$, $\omega$, $\phi$, and $B$ into the sinusoidal equation:

$y = 50 \sin\left(1.745 t - \frac{\pi}{2}\right) + 56$

This is the sinusoidal function that models the motion of the roller coaster during the run.

Do you have any questions or would you like further details?

Here are 5 related questions:

1. How would the equation change if the coaster started at the bottom of the hill?
2. What would happen to the function if the amplitude were doubled?
3. How does changing the period of the motion affect the angular frequency?
4. Can the phase shift $\phi$ be in degrees instead of radians?
5. How would this equation model the coaster if the ground level was shifted?

Tip: When solving sinusoidal problems, always ensure the correct identification of amplitude, vertical shift, and period, as these control the height and timing of the wave.