1. **Problem Statement:** We are given three problems involving distances on the Earth's surface using longitude and latitude. 2. **Given Data:** - Radius of Earth, $r = 6400$ km - $\pi = \frac{22}{7}$ --- ### Problem 13: Distance between two ships on the equator at longitudes 45°W and 45°E 3. **Formula:** Distance along the equator between two points separated by an angle $\theta$ (in degrees) is given by: $$\text{Distance} = \frac{\theta}{360} \times 2\pi r$$ 4. **Calculation:** - The difference in longitude is $45^\circ + 45^\circ = 90^\circ$ - Substitute values: $$\text{Distance} = \frac{90}{360} \times 2 \times \frac{22}{7} \times 6400$$ 5. **Simplify:** $$= \frac{1}{4} \times 2 \times \frac{22}{7} \times 6400 = \frac{1}{2} \times \frac{22}{7} \times 6400$$ $$= \frac{22}{7} \times 3200 = 22 \times \frac{3200}{7} = 22 \times 457.14 = 10057.14 \text{ km}$$ 6. **Answer:** Distance $\approx 10000$ km (to 2 significant figures) --- ### Problem 14(a): Distance between two places X and Y on the equator at longitudes 67°E and 123°E 7. **Difference in longitude:** $$123^\circ - 67^\circ = 56^\circ$$ 8. **Distance:** $$\text{Distance} = \frac{56}{360} \times 2 \times \frac{22}{7} \times 6400$$ 9. **Simplify:** $$= \frac{56}{360} \times 2 \times \frac{22}{7} \times 6400 = \frac{56}{360} \times 2 \times \frac{22}{7} \times 6400$$ $$= \frac{56}{360} \times 2 \times \frac{22}{7} \times 6400$$ Calculate stepwise: $$\frac{56}{360} = 0.1556$$ $$2 \times \frac{22}{7} = 2 \times 3.142857 = 6.2857$$ $$\text{Distance} = 0.1556 \times 6.2857 \times 6400 = 0.1556 \times 40228.57 = 6257.14 \text{ km}$$ 10. **Answer:** Distance $\approx 6300$ km (to 2 significant figures) --- ### Problem 14(b): Distance from X (67°E) to the North Pole 11. **Explanation:** - The North Pole is at latitude 90°N. - X is on the equator (latitude 0°). - Distance from equator to pole along a meridian is a quarter of Earth's circumference. 12. **Formula:** $$\text{Distance} = \frac{1}{4} \times 2\pi r = \frac{1}{4} \times 2 \times \frac{22}{7} \times 6400$$ 13. **Simplify:** $$= \frac{1}{2} \times \frac{22}{7} \times 6400 = 11 \times \frac{6400}{7} = 11 \times 914.29 = 10057.14 \text{ km}$$ 14. **Answer:** Distance $\approx 10000$ km (to 2 significant figures) --- ### Problem 15: Difference in longitude between two points P and Q 8500 km apart on the equator 15. **Formula:** $$\text{Distance} = \frac{\theta}{360} \times 2\pi r$$ 16. **Rearranged to find $\theta$:** $$\theta = \frac{\text{Distance} \times 360}{2\pi r}$$ 17. **Substitute values:** $$\theta = \frac{8500 \times 360}{2 \times \frac{22}{7} \times 6400}$$ 18. **Simplify denominator:** $$2 \times \frac{22}{7} \times 6400 = 2 \times 3.142857 \times 6400 = 40228.57$$ 19. **Calculate $\theta$:** $$\theta = \frac{8500 \times 360}{40228.57} = \frac{3060000}{40228.57} = 76.08^\circ$$ 20. **Answer:** Difference in longitude $\approx 76^\circ$ (to 2 significant figures)