1. **Problem 7:** A boat is due south of a lighthouse and sails on a bearing of 292° for 51 km until it is due west of the lighthouse. Find how far away it is from the lighthouse now. 2. **Understanding the problem:** The boat starts south of the lighthouse, sails 51 km on a bearing of 292°, and ends up due west of the lighthouse. We want the straight-line distance from the boat's final position to the lighthouse. 3. **Key concepts:** Bearing is measured clockwise from north. A bearing of 292° means the boat travels northwest, specifically 292° - 270° = 22° north of west. 4. **Set up coordinate system:** Place the lighthouse at the origin (0,0). The boat starts at (0, -d) for some distance d south (unknown but not needed). After sailing 51 km at 292°, the boat ends up due west of the lighthouse, so its final position has coordinates (-x, 0). 5. **Calculate components of the boat's movement:** The displacement vector from start to end is 51 km at 292°. - Horizontal (west-east) component: $51 \times \cos(292^\circ) = 51 \times \cos(360^\circ - 68^\circ) = 51 \times \cos(68^\circ) \approx 51 \times 0.3746 = 19.1$ km west. - Vertical (north-south) component: $51 \times \sin(292^\circ) = 51 \times \sin(360^\circ - 68^\circ) = 51 \times (-\sin(68^\circ)) \approx 51 \times (-0.9272) = -47.3$ km south. 6. **Coordinates:** Starting at (0, -d), after moving: - New x-coordinate: $0 - 19.1 = -19.1$ - New y-coordinate: $-d - 47.3$ 7. Since the boat ends due west of the lighthouse, its y-coordinate is 0: $$-d - 47.3 = 0 \implies d = -47.3$$ This means the boat started 47.3 km south of the lighthouse. 8. **Distance from lighthouse now:** The boat is at (-19.1, 0), so the distance is: $$\sqrt{(-19.1)^2 + 0^2} = 19.1 \text{ km}$$ --- 9. **Problem 8:** Wally sees an apartment block 300 m away. Angle of elevation to top is 23°, angle of depression to base is 30°. Find the height of the apartment block. 10. **Set up:** The horizontal distance from Wally to the building is 300 m. 11. **Calculate height from Wally's eye level to top:** $$h_{top} = 300 \times \tan(23^\circ) \approx 300 \times 0.4245 = 127.35 \text{ m}$$ 12. **Calculate height from Wally's eye level to base:** $$h_{base} = 300 \times \tan(30^\circ) = 300 \times 0.5774 = 173.2 \text{ m}$$ 13. **Height of apartment block:** Difference between top and base heights: $$127.35 + 173.2 = 300.55 \text{ m}$$ 14. **Final answers:** - Distance of boat from lighthouse now: $19.1$ km - Height of apartment block: $300.55$ m