1. **State the problem:** We have a Ferris wheel with height given by the equation $$h = 15 - 14 \cos(kt)$$ where $k=2.4$ and $t$ is time in seconds. We want to find the duration during one complete revolution (150 seconds) when the seat's height $h$ is at least 20 metres. 2. **Write the inequality:** We want to find $t$ such that $$h \geq 20$$ 3. **Substitute $h$:** $$15 - 14 \cos(2.4t) \geq 20$$ 4. **Isolate cosine term:** $$-14 \cos(2.4t) \geq 5$$ Divide both sides by $-14$ (note this reverses inequality): $$\cos(2.4t) \leq \frac{5}{-14} = -\frac{5}{14}$$ Intermediate step showing cancellation: $$\cos(2.4t) \leq \cancel{-\frac{5}{14}}$$ (division by $-14$ reverses inequality) 5. **Find the angles where cosine equals $-\frac{5}{14}$:** Let $$\theta = 2.4t$$ We solve $$\cos \theta = -\frac{5}{14} \approx -0.3571$$ Using inverse cosine: $$\theta_1 = \cos^{-1}(-0.3571) \approx 1.936 \text{ radians}$$ $$\theta_2 = 2\pi - 1.936 = 4.347 \text{ radians}$$ 6. **Determine intervals where $$\cos \theta \leq -\frac{5}{14}$$:** Cosine is less than or equal to $-0.3571$ between $$\theta_1$$ and $$\theta_2$$. 7. **Convert back to time $t$:** $$t = \frac{\theta}{2.4}$$ So the seat is at or above 20 m when $$t \in \left[ \frac{1.936}{2.4}, \frac{4.347}{2.4} \right] = [0.807, 1.811] \text{ seconds}$$ 8. **Calculate duration:** $$\Delta t = 1.811 - 0.807 = 1.004 \text{ seconds}$$ 9. **Adjust for one full revolution:** The Ferris wheel period is 150 seconds, and the cosine function repeats every $$\frac{2\pi}{k} = \frac{2\pi}{2.4} \approx 2.618$$ seconds. Since the problem states the seat returns to lowest point after 150 seconds, the function is scaled so that one revolution corresponds to 150 seconds, not 2.618 seconds. Therefore, the time scale is stretched by a factor: $$\text{scale} = \frac{150}{2.618} \approx 57.3$$ Multiply the duration by this scale: $$1.004 \times 57.3 = 57.5 \text{ seconds}$$ 10. **Final answer:** The seat is 20 metres or more above the ground for approximately **57.5 seconds** during one complete revolution.