1. **Problem Statement:** You are on a Ferris wheel with diameter 40 feet, taking 8 seconds for one full revolution. Your height above the ground varies sinusoidally with time $t$ seconds since the wheel started. The top of the wheel is 43 feet above the ground, and it takes 3 seconds to reach the top from the start. 2. **Known values and formulas:** - Diameter $D = 40$ feet, so radius $r = \frac{D}{2} = 20$ feet. - Period $T = 8$ seconds. - Maximum height $H_{max} = 43$ feet. - The height function is sinusoidal: $$h(t) = A \sin(B(t - C)) + D$$ where: - $A$ is amplitude, - $B = \frac{2\pi}{T}$ is angular frequency, - $C$ is horizontal shift (phase shift), - $D$ is vertical shift (midline). 3. **Find amplitude $A$ and vertical shift $D$:** - The amplitude is half the distance between max and min heights. - The top is 43 feet, so the center height is $D = 43 - r = 43 - 20 = 23$ feet. - Amplitude $A = r = 20$ feet. 4. **Find angular frequency $B$:** $$B = \frac{2\pi}{T} = \frac{2\pi}{8} = \frac{\pi}{4}$$ 5. **Find phase shift $C$:** - The wheel starts at the bottom (lowest point) at $t=0$. - The lowest height is $D - A = 23 - 20 = 3$ feet. - The height reaches maximum at $t=3$ seconds. - Since sine reaches max at $\frac{\pi}{2}$, solve: $$B(t - C) = \frac{\pi}{2}$$ $$\frac{\pi}{4}(3 - C) = \frac{\pi}{2}$$ Divide both sides by $\frac{\pi}{4}$: $$\cancel{\frac{\pi}{4}}(3 - C) = \cancel{\frac{\pi}{4}} 2$$ $$3 - C = 2$$ $$C = 1$$ 6. **Write the height function:** $$h(t) = 20 \sin\left(\frac{\pi}{4}(t - 1)\right) + 23$$ 7. **Lowest height and why it is greater than zero:** - Lowest height is $D - A = 23 - 20 = 3$ feet. - This is greater than zero because the Ferris wheel is elevated 3 feet above the ground (the center is 23 feet, not at ground level). 8. **Predict height at any time $t$ using the equation above.** **Final answer:** $$\boxed{h(t) = 20 \sin\left(\frac{\pi}{4}(t - 1)\right) + 23}$$