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Two towns Q and R with coordinates Q(55°N, 35°W) and R(55°N, 145°E) lie on the earth's surface. (a) Find the distance in nautical miles between Q and R along: (i) the longitude; (ii) the latitude.

Mathematics

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Haya Shamah, twende kazi na swali hili la urambazaji.

22. Miji miwili Q na R yenye kuratibu Q(55∘N,35∘W)Q(55^\circ N, 35^\circ W) na R(55∘N,145∘E)R(55^\circ N, 145^\circ E) iko kwenye uso wa dunia.

Sehemu (a): Tafuta umbali katika maili za baharini kati ya Q na R kando ya:

Step 1: Tambua kuratibu na tofauti ya longitudo.
Kuratibu za Q: 55∘N,35∘W55^\circ N, 35^\circ W
Kuratibu za R: 55∘N,145∘E55^\circ N, 145^\circ E
Latitudo ni sawa: ϕ=55∘N\phi = 55^\circ N.
Tofauti ya longitudo: Δλ=35∘+145∘=180∘\Delta \lambda = 35^\circ + 145^\circ = 180^\circ.

Part i

longitudo

Hii inamaanisha umbali wa duara kuu (great circle distance) unaopitia ncha za dunia. Kwa kuwa Q na R ziko kwenye latitudo moja na tofauti ya longitudo ni 180∘180^\circ, njia fupi zaidi kupitia longitudo ni kupitia Ncha ya Kaskazini au Kusini.
Umbali kutoka Q hadi Ncha ya Kaskazini:
Angle = 90∘−55∘=35∘90^\circ - 55^\circ = 35^\circ.
Umbali = 60×35∘=210060 \times 35^\circ = 2100 maili za baharini.
Umbali kutoka Ncha ya Kaskazini hadi R:
Angle = 90∘−55∘=35∘90^\circ - 55^\circ = 35^\circ.
Umbali = 60×35∘=210060 \times 35^\circ = 2100 maili za baharini.
Jumla ya umbali kando ya longitudo (kupitia ncha) ni:

2100+2100=4200 maili za baharini2100 + 2100 = 4200 \text{ maili za baharini}

Umbali kando ya longitudo ni 4200 maili za baharini\boxed{\mathbf{4200 \text{ maili za baharini}}}.

Part ii· 3 steps

latitudo

Hii inamaanisha umbali kando ya sambamba ya latitudo 55∘N55^\circ N.
Tumia fomula: Umbali = 60×Δλ×cos⁡ϕ60 \times \Delta \lambda \times \cos \phi.

Umbali=60×180∘×cos⁡55∘Umbali = 60 \times 180^\circ \times \cos 55^\circ Umbali=10800×0.573576Umbali = 10800 \times 0.573576 Umbali=6194.6255 maili za bahariniUmbali = 6194.6255 \text{ maili za baharini}

Umbali kando ya latitudo ni 6194.63 maili za baharini\boxed{\mathbf{6194.63 \text{ maili za baharini}}}.

Sehemu (b): Siku ya Jumatatu saa 1300 h saa za huko, ndege mbili A na B kila moja ikiruka kwa kasi ya mafundo 250 zilianza wakati mmoja kutoka Q kwenda R. Ndege A inaruka kando ya longitudo wakati B inaruka kando ya latitudo. Kokotoa muda:

  1. Kokotoa muda wa ndege A kutua R.

    Kasi = 250 mafundo (maili za baharini kwa saa).
    Umbali wa ndege A (kando ya longitudo) = 4200 maili za baharini (kutoka (a)(i)).
    Muda wa safari = UmbaliKasi=4200250=16.8\frac{\text{Umbali}}{\text{Kasi}} = \frac{4200}{250} = 16.8 masaa.

    0.8 masaa =0.8×60=48= 0.8 \times 60 = 48 dakika.
    Muda wa safari = 16 masaa na 48 dakika.

    Muda wa kuanza (saa za Q): Jumatatu 13:00.
    Muda wa kufika (saa za Q): Jumatatu 13:00 + 16 masaa 48 dakika = Jumanne 05:48.

    Tofauti ya saa kati ya Q na R:
    Tofauti ya longitudo = 180∘180^\circ.
    1∘1^\circ ya longitudo = 4 dakika.
    Tofauti ya saa = 180∘×4 min/degree=720180^\circ \times 4 \text{ min/degree} = 720 dakika = 12 masaa.
    Kwa kuwa R iko Mashariki mwa Q, saa za R ziko mbele kwa masaa 12.
    Muda wa kufika (saa za R): Jumanne 05:48 + 12 masaa = Jumanne 17:48.
    Ndege A ilitua R saa Jumanne 17:48\boxed{\mathbf{Jumanne \ 17:48}}.

  2. Kokotoa muda wa ndege B kutua R.

    Umbali wa ndege B (kando ya latitudo) = 6194.63 maili za baharini (kutoka (a)(ii)).
    Muda wa safari = UmbaliKasi=6194.63250=24.77852\frac{\text{Umbali}}{\text{Kasi}} = \frac{6194.63}{250} = 24.77852 masaa.
    0.77852 masaa =0.77852×60=46.71= 0.77852 \times 60 = 46.71 dakika.
    Muda wa safari = 24 masaa na 47 dakika (ukizungusha).

    Muda wa kuanza (saa za Q): Jumatatu 13:00.
    Muda wa kufika (saa za Q): Jumatatu 13:00 + 24 masaa 47 dakika = Jumanne 13:47.
    Muda wa kufika (saa za R): Jumanne 13:47 + 12 masaa = Jumatano 01:47.
    Ndege B ilitua R saa Jumatano 01:47\boxed{\mathbf{Jumatano \ 01:47}}.

  3. Kokotoa tofauti ya muda wa kutua.

    Muda wa kutua ndege A: Jumanne 17:48 (saa za R).
    Muda wa kutua ndege B: Jumatano 01:47 (saa za R).
    Tofauti ya muda = (Jumatano 01:47) - (Jumanne 17:48).
    Kuanzia Jumanne 17:48 hadi Jumanne 24:00 (usiku wa manane) ni 24:00−17:48=624:00 - 17:48 = 6 masaa 12 dakika.
    Kuanzia Jumanne 24:00 hadi Jumatano 01:47 ni 1 saa 47 dakika.
    Jumla ya tofauti = 6 masaa 12 dakika + 1 saa 47 dakika = 7 masaa 59 dakika.
    Tofauti ya muda wa kutua ni 7 masaa 59 dakika\boxed{\mathbf{7 \text{ masaa } 59 \text{ dakika}}}.

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