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.

To model the height of the red peg as a function of time using a sine function, we need to establish the amplitude, midline, and period based on the given information. The midline (the average height) is calculated as \((0.5 + 3.5) / 2 = 2.0\) m. The amplitude (half the distance between the maximum and minimum heights) is \((3.5 - 0.5) / 2 = 1.5\) m. The period is given as \(3\pi\) seconds, so we can find the angular frequency, which is \(\frac{2\pi}{\text{period}} = \frac{2\pi}{3\pi} = \frac{2}{3}\). Therefore, the general form of the sine function can be represented as: \[ h(t) = A \sin(B(t - C)) + D \] Where: - \(A = 1.5\) m (amplitude) - \(B = \frac{2}{3}\) (angular frequency) - \(C = 0\) (horizontal shift) - \(D = 2.0\) m (midline) This gives us the final equation: \[ h(t) = 1.5 \sin\left(\frac{2}{3}t\right) + 2 \] Now, let's visualize this! Imagine a vertical axis for height (0 to 4 m) and a horizontal axis for time (0 to \(3\pi\) seconds). The sine wave oscillates between 0.5 m and 3.5 m, crossing the midline at \(2.0\) m. The peaks occur at \(t = \frac{3\pi}{4}\) and \(t = \frac{9\pi}{4}\), with the corresponding heights reaching maximum and minimum values at the respective intervals. To keep the illustration clear, label the points accordingly with lines showing the amplitude and midline.