1. **Problem statement:** We want to analyze the swing described as a pendulum with arm length $22$ m, swinging up to an angle of $120^\circ$, reaching a maximum height of $45$ m. 2. **Known data:** - Length of swing arm $L = 22$ m - Maximum angle $\theta = 120^\circ$ - Maximum height reached $h = 45$ m 3. **Goal:** Verify the maximum height reached by the swing using the pendulum arm length and angle. 4. **Formula:** The vertical height $h$ gained by the pendulum bob from the lowest point is given by: $$h = L - L \cos(\theta) = L(1 - \cos(\theta))$$ where $\theta$ is the angle from the vertical. 5. **Important note:** The angle $120^\circ$ is likely the total swing angle from one side to the other, so the angle from vertical to the highest point is half of that: $$\alpha = \frac{120^\circ}{2} = 60^\circ$$ 6. **Calculate height:** $$h = 22 \times (1 - \cos(60^\circ))$$ Since $\cos(60^\circ) = 0.5$, $$h = 22 \times (1 - 0.5) = 22 \times 0.5 = 11 \text{ m}$$ 7. **Interpretation:** The height gained from the lowest point is $11$ m, so if the lowest point is at ground level, the top of the swing reaches $11$ m above that. 8. **Given maximum height is 45 m:** This suggests the base of the swing is elevated or the height is measured from a different reference point. 9. **Horizontal distance check:** The horizontal distance covered at max angle is: $$x = L \sin(\alpha) = 22 \times \sin(60^\circ) = 22 \times \frac{\sqrt{3}}{2} \approx 19.05 \text{ m}$$ 10. **Summary:** - Swing arm length: 22 m - Max angle from vertical: 60° - Height gained from lowest point: 11 m - Horizontal distance at max angle: approx. 19.05 m The given maximum height of 45 m likely includes the height of the pivot point above ground. **Final answer:** The height gained by the swing from the lowest point at $60^\circ$ is $11$ m.