1. **State the problem:** We need to write an equation for the height $h = f(t)$ of a person on a ferris wheel as a function of time $t$ in minutes. 2. **Given information:** - Diameter of ferris wheel = 10 meters, so radius $r = \frac{10}{2} = 5$ meters. - The platform is 1 meter above the ground. - The wheel completes 1 full revolution in 14 minutes. - Boarding is at the three o'clock position, which corresponds to the lowest point on the wheel. 3. **Formula and explanation:** The height function for a ferris wheel can be modeled by a sine function: $$h(t) = A \sin(B(t - C)) + D$$ where: - $A$ is the amplitude (radius of the wheel), - $B = \frac{2\pi}{T}$ where $T$ is the period (time for one revolution), - $C$ is the horizontal shift (phase shift), - $D$ is the vertical shift (height of the center of the wheel above the ground). 4. **Calculate parameters:** - Amplitude $A = 5$ - Period $T = 14$ minutes, so $$B = \frac{2\pi}{14} = \frac{\pi}{7}$$ - Vertical shift $D$ is the height of the center of the wheel. Since the wheel radius is 5 and the platform is 1 meter above ground, the center is at height: $$D = 5 + 1 = 6$$ 5. **Determine phase shift $C$:** - Boarding at three o'clock means starting at the lowest point of the wheel. - The sine function normally starts at zero height at $t=0$. - The lowest point corresponds to the minimum of the sine function, which occurs at $\sin(\frac{3\pi}{2}) = -1$. - To model this, we shift the sine function so that at $t=0$, the argument equals $\frac{3\pi}{2}$: $$B(t - C) = \frac{3\pi}{2} \quad \Rightarrow \quad -B C = \frac{3\pi}{2} \quad \Rightarrow \quad C = -\frac{3\pi}{2B}$$ - Substitute $B = \frac{\pi}{7}$: $$C = -\frac{3\pi/2}{\pi/7} = -\frac{3\pi}{2} \times \frac{7}{\pi} = -\frac{21}{2} = -10.5$$ 6. **Write the final function:** $$h(t) = 5 \sin\left(\frac{\pi}{7}(t + 10.5)\right) + 6$$ This function gives the height in meters above the ground at time $t$ minutes after the wheel starts turning, starting from the lowest point at $t=0$. **Final answer:** $$\boxed{h(t) = 5 \sin\left(\frac{\pi}{7}(t + 10.5)\right) + 6}$$