1. **Problem statement:** We need to find a sine function that models the height $h$ of a Ferris wheel car above the ground as a function of time $t$. 2. **Given data:** - Diameter of Ferris wheel = 18.5 m - Angular velocity = 0.2 rad/s - Height of lowest car above ground = 3 m 3. **Step 1: Understand the motion and parameters** - Radius $r = \frac{18.5}{2} = 9.25$ m - The Ferris wheel rotates at angular velocity $\omega = 0.2$ rad/s - The height of the car varies sinusoidally with time because it moves in a circle. 4. **Step 2: General sine function for height** The height $h(t)$ can be modeled as: $$h(t) = A \sin(\omega t + \phi) + D$$ where: - $A$ is the amplitude (radius of the wheel) - $\omega$ is the angular velocity - $\phi$ is the phase shift - $D$ is the vertical shift (center height) 5. **Step 3: Determine amplitude $A$ and vertical shift $D$** - Amplitude $A = 9.25$ m (radius) - The center of the wheel is at height $r + 3 = 9.25 + 3 = 12.25$ m above ground, so $D = 12.25$ 6. **Step 4: Determine phase shift $\phi$** - At $t=0$, the car is at the lowest point, so height $h(0) = 3$ m. - Using the sine function: $$h(0) = 9.25 \sin(\phi) + 12.25 = 3$$ $$9.25 \sin(\phi) = 3 - 12.25 = -9.25$$ $$\sin(\phi) = -1$$ - This means $\phi = -\frac{\pi}{2}$ or $\frac{3\pi}{2}$ (choosing $-\frac{\pi}{2}$ for simplicity). 7. **Step 5: Write the final function** $$h(t) = 9.25 \sin(0.2 t - \frac{\pi}{2}) + 12.25$$ **Answer:** The height of the car above the ground as a function of time is $$h(t) = 9.25 \sin\left(0.2 t - \frac{\pi}{2}\right) + 12.25$$ meters.