Calculate the distance btwn the points P(40°N, 15°W) and Q(40°N, 160°W) in: (a) km (b)nm

$$D = 0.402777... \times 2\pi (6370 \text{ km}) \times 0.766044...$$ $$D \approx 12340.83 \text{ km}$$ $$D \approx \boxed{\text{12341 km}}$$

b) Katika nm:

$$D_{nm} = \frac{D_{km}}{1.852 \text{ km/nm}}$$ $$D_{nm} = \frac{12340.83 \text{ km}}{1.852 \text{ km/nm}}$$ $$D_{nm} \approx 6663.51 \text{ nm}$$ $$D_{nm} \approx \boxed{\text{6664 nm}}$$

2) Tafuta mzingo wa latitudo $70^\circ S$.

Latitudo ni $\alpha = 70^\circ$. Fomula ya mzingo wa latitudo ni:

$$C = 2\pi R \cos \alpha$$

a) Katika km:

$$C = 2\pi (6370 \text{ km}) \times 0.342020...$$ $$C \approx 13680.84 \text{ km}$$ $$C \approx \boxed{\text{13681 km}}$$

b) Katika nm:

$$C_{nm} = \frac{C_{km}}{1.852 \text{ km/nm}}$$ $$C_{nm} = \frac{13680.84 \text{ km}}{1.852 \text{ km/nm}}$$ $$C_{nm} \approx 7387.06 \text{ nm}$$ $$C_{nm} \approx \boxed{\text{7387 nm}}$$

3) Tafuta mzingo wa Tropiki ya Kansa ($23\frac{1}{2}^\circ N$).

Latitudo ni $\alpha = 23.5^\circ$. Fomula ya mzingo wa latitudo ni:

$$C = 2\pi R \cos \alpha$$

a) Katika km:

$$C = 2\pi (6370 \text{ km}) \times 0.917063...$$ $$C \approx 36706.70 \text{ km}$$ $$C \approx \boxed{\text{36707 km}}$$

b) Katika nm:

$$C_{nm} = \frac{C_{km}}{1.852 \text{ km/nm}}$$ $$C_{nm} = \frac{36706.70 \text{ km}}{1.852 \text{ km/nm}}$$ $$C_{nm} \approx 19819.06 \text{ nm}$$ $$C_{nm} \approx \boxed{\text{19819 nm}}$$

4) Tafuta urefu wa sambamba kupitia ($1^\circ S, 37^\circ E$).

Hii inamaanisha mzingo kamili wa latitudo $1^\circ S$. Latitudo ni $\alpha = 1^\circ$. Fomula ya mzingo wa latitudo ni:

$$C = 2\pi R \cos \alpha$$

a) Katika km:

$$C = 2\pi (6370 \text{ km}) \times 0.999847...$$ $$C \approx 40023.80 \text{ km}$$ $$C \approx \boxed{\text{40024 km}}$$

b) Katika nm:

$$C_{nm} = \frac{C_{km}}{1.852 \text{ km/nm}}$$ $$C_{nm} = \frac{40023.80 \text{ km}}{1.852 \text{ km/nm}}$$ $$C_{nm} \approx 21611.12 \text{ nm}$$ $$C_{nm} \approx \boxed{\text{21611 nm}}$$