To model the height of a passenger on the Ferris wheel as a function of time, we can use a sinusoidal function. Here's the step-by-step process to derive the equation:

1. Determine the angular velocity:

   • The Ferris wheel makes 5 revolutions every 2 minutes.
   • Convert this to revolutions per second: $\frac{5 \text{ revolutions}}{120 \text{ seconds}} = \frac{1}{24} \text{ revolutions per second}$.
   • Convert revolutions per second to angular velocity in radians per second: $\frac{1}{24} \times 2\pi = \frac{\pi}{12} \text{ radians per second}$.

2. Determine the amplitude:

   • The diameter of the Ferris wheel is 20 meters, so the radius is $10$ meters. The amplitude $A$ of the sinusoidal function is $10$ meters.

3. Determine the vertical shift:

   • At the bottom of the ride, the passenger is 3 meters above the ground. Since the radius is 10 meters, the center of the Ferris wheel is $10 + 3 = 13$ meters above the ground. Hence, the vertical shift $D$ is 13 meters.

4. Determine the phase shift:

   • We need the function to start at the bottom of the Ferris wheel. A cosine function typically starts at its maximum, so we need to shift it. However, if we use a sine function, it starts at the middle going up, which is more intuitive here as it simplifies our model.

Thus, the height $h(t)$ above the ground can be modeled by the equation: $h(t) = 10 \sin\left(\frac{\pi}{12} t - \frac{\pi}{2}\right) + 13$

To find the height at $t = 14$ seconds:

1. Plug $t = 14$ into the equation: $h(14) = 10 \sin\left(\frac{\pi}{12} \cdot 14 - \frac{\pi}{2}\right) + 13$

2. Simplify inside the sine function: $\frac{\pi}{12} \cdot 14 = \frac{14\pi}{12} = \frac{7\pi}{6}$ $\frac{7\pi}{6} - \frac{\pi}{2} = \frac{7\pi}{6} - \frac{3\pi}{6} = \frac{4\pi}{6} = \frac{2\pi}{3}$

3. Find the sine value: $\sin\left(\frac{2\pi}{3}\right) = \sin\left(120^\circ\right) = \sin\left(60^\circ\right) = \frac{\sqrt{3}}{2}$

4. Substitute back into the equation: $h(14) = 10 \cdot \frac{\sqrt{3}}{2} + 13 = 5\sqrt{3} + 13$

Thus, the exact height of the passenger 14 seconds into their ride is $5\sqrt{3} + 13$ meters.

Would you like more details on any part of this solution, or do you have any specific questions?

Here are some other questions you might be interested in:

1. How do you determine the period of a sinusoidal function?
2. How do you convert angular velocity from revolutions per minute to radians per second?
3. What are some real-world applications of sinusoidal functions?
4. How do you solve for the phase shift in a sinusoidal function?
5. How can you graph the height function of the Ferris wheel?

Tip: When modeling real-world scenarios with sinusoidal functions, always consider the amplitude, vertical shift, and phase shift to accurately represent the scenario.