3) contestant on a game show spins a wheel located on a plane perpendicular to the floor. He grabs the only red peg on the circumference of the wheel which is 2 m above the floor and pushes it downward. The red peg reaches a minimum height of 0.5 m above the floor and a maximum of 3.5 m above the floor. If one cycle is completed in $$ 3 \pi $$ seconds, determine the equation of the sine function and model your solution with a clear, labelled sketch.

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It seems that I am unable to generate a plot directly. However, I can guide you on how to sketch the function based on the derived equation.

### Instructions for Sketching the Function

1. **Draw the Axes**:
   - Label the horizontal axis as "Time (s)" and the vertical axis as "Height (m)".
   - Mark the vertical axis from $$ 0 $$ m to $$ 4 $$ m.

2. **Identify Key Points**:
   - Start at $$ (0, 2) $$ m (the midline).
   - At $$ t = \frac{3\pi}{4} $$ (approximately $$ 2.36 $$ s), the height reaches its maximum of $$ 3.5 $$ m.
   - At $$ t = \frac{3\pi}{2} $$ (approximately $$ 4.71 $$ s), the height returns to $$ 2 $$ m.
   - At $$ t = \frac{9\pi}{4} $$ (approximately $$ 7.85 $$ s), the height reaches its minimum of $$ 0.5 $$ m.
   - Finally, at $$ t = 3\pi $$ (approximately $$ 9.42 $$ s), the height returns to $$ 2 $$ m.

3. **Plot the Points**:
   - Mark the points on the graph:
     - $$ (0, 2) $$
     - $$ \left(\frac{3\pi}{4}, 3.5\right) $$
     - $$ \left(\frac{3\pi}{2}, 2\right) $$
     - $$ \left(\frac{9\pi}{4}, 0.5\right) $$
     - $$ (3\pi, 2) $$

4. **Draw the Sine Wave**:
   - Connect the points smoothly to form a sine wave that oscillates between $$ 0.5 $$ m and $$ 3.5 $$ m, with the midline at $$ 2 $$ m.

5. **Label the Graph**:
   - Indicate the maximum and minimum heights on the graph.
   - Optionally, you can label the period of the function as $$ 3\pi $$ seconds.

This sketch will visually represent the height of the red peg over time as it oscillates due to the spinning wheel. If you have graphing software or a graphing calculator, you can input the equation $$ h(t) = 1.5 \sin\left(\frac{2}{3} t\right) + 2 $$ to generate a more precise graph.
The height of the red peg can be modeled by the sine function: $$ h(t) = 1.5 \sin\left(\frac{2}{3} t\right) + 2 $$ This function oscillates between 0.5 m and 3.5 m with a period of $$ 3\pi $$ seconds.

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