To solve this problem, we will use the Earth's radius $R = 6370 \text{ km}$ and $\pi \approx 3.14159$. i) Length of the circle of latitude $52^\circ N$ The formula for the length of a circle of latitude is $L = 2\pi R \cos \phi$, where $R$ is the Earth's radius and $\phi$ is the latitude. Given latitude $\phi = 52^\circ N$. Step 1: Substitute the values into the formula. $$L = 2 \times \pi \times 6370 \text{ km} \times \cos(52^\circ)$$ Step 2: Calculate the value. $$L = 2 \times 3.14159265 \times 6370 \times 0.61566147$$ $$L = 24690.69 \text{ km}$$ Step 3: Round to a suitable number of significant figures (e.g., 4 significant figures). $$L \approx \boxed{24690 \text{ km}}$$ ii) Distance BC along the longitude $76^\circ E$ Points B and C are on the same longitude ($76^\circ E$). Latitude of B is $52^\circ N$. Latitude of C is $78^\circ S$. Since the latitudes are in different hemispheres (North and South), the angular difference $\theta$ is the sum of their absolute values. Step 1: Calculate the angular difference $\theta$. $$\theta = 52^\circ + 78^\circ = 130^\circ$$ Step 2: The distance along a longitude (which is a great circle) is given by $D = \frac{\theta}{360^\circ} \times 2\pi R$. Substitute the values into the formula. $$D_{BC} = \frac{130^\circ}{360^\circ} \times 2 \times \pi \times 6370 \text{ km}$$ Step 3: Calculate the value. $$D_{BC} = \frac{13}{36} \times 2 \times 3.14159265 \times 6370$$ $$D_{BC} = 0.36111111 \times 40023.8946$$ $$D_{BC} = 14453.70 \text{ km}$$ Step 4: Round to a suitable number of significant figures (e.g., 4 significant figures). $$D_{BC} \approx \boxed{14450 \text{ km}}$$ iii) Distance DC along the circle of latitude $78^\circ S$ Points D and C are on the same circle of latitude ($78^\circ S$). Longitude of D is $43^\circ W$. Longitude of C is $76^\circ E$. Since the longitudes are in different hemispheres (West and East), the angular difference $\alpha$ is the sum of their absolute values. Step 1: Calculate the angular difference $\alpha$. $$\alpha = 43^\circ + 76^\circ = 119^\circ$$ Step 2: The distance along a circle of latitude is given by $D = \frac{\alpha}{360^\circ} \times 2\pi R \cos \phi$. Given latitude $\phi = 78^\circ S$. Substitute the values into the formula. $$D_{DC} = \frac{119^\circ}{360^\circ} \times 2 \times \pi \times 6370 \text{ km} \times \cos(78^\circ)$$ Step 3: Calculate the value. $$D_{DC} = \frac{119}{360} \times 2 \times 3.14159265 \times 6370 \times 0.20791169$$ $$D_{DC} = 0.33055555 \times 40023.8946 \times 0.20791169$$ $$D_{DC} = 0.33055555 \times 8321.030$$ $$D_{DC} = 2751.00 \text{ km}$$ Step 4: Round to a suitable number of significant figures (e.g., 4 significant figures). $$D_{DC} \approx \boxed{2751 \text{ km}}$$