A ferris wheel at an amusement park has a diameter of 20 m and travels at a rate of 5 revolutions every 2 minutes.  At the bottom of the ride, the passenger is 3 m above the ground.  Write an equation to represent the height, h, in metres, above the ground at time t, in seconds. Determine the exact height of the passenger 14 seconds into their ride.

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.

Learn how to model the height of a passenger on a Ferris wheel using a sinusoidal function. This example covers the derivation of the height function based on the wheel's diameter, speed, and initial height above ground. Suitable for Grades 11-12.

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