Find the distance between the points P(35°N, 50°E) and (26° 30'S, 50°E) on the earth's surface in: (a) km (b) nm-

Tutatumia radius ya Dunia $R = 6370$ km na kwamba 1 nautical mile (nm) $= 1.852$ km.

5) Tafuta umbali katika nautical miles kati ya pointi A($45^\circ N, 35^\circ W$) na B($45^\circ N, 100^\circ E$) kando ya sambamba ya latitudo.

Pointi zote mbili ziko kwenye latitudo moja ($\alpha = 45^\circ N$). Tofauti ya longitudo ($\theta$) itakuwa:

$$\theta = 35^\circ + 100^\circ = 135^\circ$$

Fomula ya umbali kando ya latitudo ni:

$$D = \frac{\theta}{360^\circ} \times 2\pi R \cos \alpha$$

Step 1: Badilisha maadili kwenye fomula ili kupata umbali katika km.

$$D_{km} = \frac{135^\circ}{360^\circ} \times 2\pi (6370 \text{ km}) \cos(45^\circ)$$

Step 2: Kokotoa.

$$D_{km} = 0.375 \times 2\pi (6370 \text{ km}) \times 0.707106...$$ $$D_{km} \approx 10609.95 \text{ km}$$

Step 3: Badilisha umbali kutoka km kwenda nm.

$$D_{nm} = \frac{D_{km}}{1.852 \text{ km/nm}}$$ $$D_{nm} = \frac{10609.95 \text{ km}}{1.852 \text{ km/nm}}$$ $$D_{nm} \approx 5728.97 \text{ nm}$$ $$D_{nm} \approx \boxed{\text{5729 nm}}$$

6) Tafuta latitudo kusini mwa ikweta ambapo unaposafiri kando ya latitudo kwa 80 nm, mabadiliko ya $2^\circ$ katika longitudo hufanywa.

Tunapewa umbali $D = 80$ nm na tofauti ya longitudo $\theta = 2^\circ$. Tunahitaji kupata latitudo $\alpha$.
Fomula ya umbali kando ya latitudo ni:

$$D = \frac{\theta}{360^\circ} \times 2\pi R \cos \alpha$$

Step 1: Badilisha umbali kutoka nm kwenda km.

$$D_{km} = 80 \text{ nm} \times 1.852 \text{ km/nm} = 148.16 \text{ km}$$

Step 2: Panga upya fomula ili kupata $\cos \alpha$.

$$\cos \alpha = \frac{D_{km} \times 360^\circ}{2\pi R \theta}$$

Step 3: Badilisha maadili na kokotoa $\cos \alpha$.

$$\cos \alpha = \frac{148.16 \text{ km} \times 360^\circ}{2\pi (6370 \text{ km}) \times 2^\circ}$$ $$\cos \alpha = \frac{53337.6}{80048.66...}$$ $$\cos \alpha \approx 0.66631$$

Step 4: Tafuta $\alpha$ kwa kutumia inverse cosine.

$$\alpha = \cos^{-1}(0.66631)$$ $$\alpha \approx 48.21^\circ$$

Kwa kuwa latitudo iko kusini mwa ikweta, jibu ni:

$$\alpha \approx \boxed{48.21^\circ S}$$

7) Tafuta umbali kati ya pointi P($35^\circ N, 50^\circ E$) na Q($26^\circ 30' S, 50^\circ E$) kwenye uso wa Dunia.

Pointi zote mbili ziko kwenye longitudo moja ($50^\circ E$). Hii inamaanisha umbali utapimwa kando ya meridian (duara kuu).
Latitudo za pointi ni $35^\circ N$ na $26^\circ 30' S$.
Tofauti ya latitudo ($\phi$) itakuwa jumla ya latitudo kwa sababu ziko pande tofauti za ikweta.

$$\phi = 35^\circ + 26^\circ 30' = 61^\circ 30'$$

Badilisha dakika kuwa digrii: $30' = 0.5^\circ$.

$$\phi = 61.5^\circ$$

Fomula ya umbali kando ya duara kuu ni:

$$D = \frac{\phi}{360^\circ} \times 2\pi R$$