1. **Problem statement:** From an aeroplane at a horizontal distance of 1050 m from the base of a control tower, the angles of depression to the top and base of the tower are 36° and 41° respectively. We need to find: (i) The height of the control tower. (ii) The shortest distance between the aeroplane and the base of the control tower. 2. **Diagram and variables:** Let the height of the aeroplane above the ground be $h_a$. Let the height of the control tower be $h_t$. The horizontal distance from the aeroplane to the base of the tower is $d = 1050$ m. 3. **Using trigonometry:** The angle of depression to the top of the tower is 36°, so the angle between the horizontal line from the plane and the line of sight to the top is 36°. The angle of depression to the base of the tower is 41°. 4. **Formulas:** From the right triangle formed by the aeroplane, the top of the tower, and the horizontal line: $$\tan(36^\circ) = \frac{h_a - h_t}{d}$$ From the right triangle formed by the aeroplane, the base of the tower, and the horizontal line: $$\tan(41^\circ) = \frac{h_a}{d}$$ 5. **Calculate $h_a$ from the base angle:** $$h_a = d \times \tan(41^\circ) = 1050 \times \tan(41^\circ)$$ Calculate $\tan(41^\circ)$: $$\tan(41^\circ) \approx 0.8693$$ So, $$h_a = 1050 \times 0.8693 = 912.765$$ 6. **Calculate $h_t$ from the top angle:** $$\tan(36^\circ) = \frac{h_a - h_t}{d} \Rightarrow h_t = h_a - d \times \tan(36^\circ)$$ Calculate $\tan(36^\circ)$: $$\tan(36^\circ) \approx 0.7265$$ So, $$h_t = 912.765 - 1050 \times 0.7265 = 912.765 - 762.825 = 149.94$$ 7. **Height of the control tower:** $$\boxed{150 \text{ m (nearest metre)}}$$ 8. **Shortest distance between the aeroplane and the base of the tower:** This is the straight line distance from the aeroplane to the base, which is the hypotenuse of the right triangle with legs $h_a$ and $d$: $$\text{Distance} = \sqrt{d^2 + h_a^2} = \sqrt{1050^2 + 912.765^2}$$ Calculate: $$1050^2 = 1,102,500$$ $$912.765^2 \approx 833,191.5$$ Sum: $$1,102,500 + 833,191.5 = 1,935,691.5$$ Square root: $$\sqrt{1,935,691.5} \approx 1,391.3$$ 9. **Shortest distance:** $$\boxed{1391 \text{ m (nearest metre)}}$$