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Two airliners, A and B, fly from point P to points Q and R respectively. Given their coordinates and speed, determine which airliner arrives earlier and by how much.

Mathematics

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Answer

Airliner B arrives earlier by 167 minutes\text{Airliner B arrives earlier by 167 minutes}

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Here's the solution to question 30.

The problem asks us to determine which airliner arrives at its destination earlier and by how long, given their coordinates and speeds.

Given:
Towns: P(60∘N,35∘W)P(60^\circ N, 35^\circ W), Q(75∘N,145∘E)Q(75^\circ N, 145^\circ E), R(60∘N,25∘E)R(60^\circ N, 25^\circ E).
Airliner A: Flies from PP to QQ using the shortest route possible. Speed is 600 km/hr600 \, \text{km/hr}.
Airliner B: Takes off from BB (which is PP) at the same time as A and flies to town RR along the parallel of latitude. Speed is 600 km/hr600 \, \text{km/hr}.

We need to calculate the time taken by each airliner. The radius of the Earth is approximately RE=6370 kmR_E = 6370 \, \text{km}.

  1. Calculate the distance for Airliner A (shortest route).

    The shortest route between two points on Earth is along a great circle. The angle θ\theta between two points (L1,λ1)(L_1, \lambda_1) and (L2,λ2)(L_2, \lambda_2) on a great circle is given by:

    cos⁡θ=sin⁡L1sin⁡L2+cos⁡L1cos⁡L2cos⁡(λ2−λ1)\cos \theta = \sin L_1 \sin L_2 + \cos L_1 \cos L_2 \cos(\lambda_2 - \lambda_1)

    For Airliner A, P(60∘N,35∘W)P(60^\circ N, 35^\circ W) and Q(75∘N,145∘E)Q(75^\circ N, 145^\circ E).
    L1=60∘L_1 = 60^\circ, L2=75∘L_2 = 75^\circ.
    λ1=−35∘\lambda_1 = -35^\circ (West), λ2=145∘\lambda_2 = 145^\circ (East).
    The difference in longitude is Δλ=λ2−λ1=145∘−(−35∘)=145∘+35∘=180∘\Delta \lambda = \lambda_2 - \lambda_1 = 145^\circ - (-35^\circ) = 145^\circ + 35^\circ = 180^\circ.

    cos⁡θ=sin⁡60∘sin⁡75∘+cos⁡60∘cos⁡75∘cos⁡180∘\cos \theta = \sin 60^\circ \sin 75^\circ + \cos 60^\circ \cos 75^\circ \cos 180^\circ cos⁡θ=(32)(6+24)+(12)(6−24)(−1)\cos \theta = \left(\frac{\sqrt{3}}{2}\right) \left(\frac{\sqrt{6}+\sqrt{2}}{4}\right) + \left(\frac{1}{2}\right) \left(\frac{\sqrt{6}-\sqrt{2}}{4}\right) (-1) cos⁡θ=18+68−6−28\cos \theta = \frac{\sqrt{18}+\sqrt{6}}{8} - \frac{\sqrt{6}-\sqrt{2}}{8} cos⁡θ=32+6−6+28=428=22\cos \theta = \frac{3\sqrt{2}+\sqrt{6}-\sqrt{6}+\sqrt{2}}{8} = \frac{4\sqrt{2}}{8} = \frac{\sqrt{2}}{2}

    So, θ=arccos⁡(22)=45∘\theta = \arccos\left(\frac{\sqrt{2}}{2}\right) = 45^\circ.
    Convert θ\theta to radians: θ=45∘×π180∘=π4\theta = 45^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{4} radians.
    The distance DA=REθD_A = R_E \theta.

    DA=6370 km×π4D_A = 6370 \, \text{km} \times \frac{\pi}{4} DA≈6370×0.785398≈5002.8 kmD_A \approx 6370 \times 0.785398 \approx 5002.8 \, \text{km}

    Time taken by Airliner A: TA=DASpeedT_A = \frac{D_A}{\text{Speed}}.

    TA=5002.8 km600 km/hr≈8.338 hoursT_A = \frac{5002.8 \, \text{km}}{600 \, \text{km/hr}} \approx 8.338 \, \text{hours}
  2. Calculate the distance for Airliner B (along the parallel of latitude).

    Airliner B flies from P(60∘N,35∘W)P(60^\circ N, 35^\circ W) to R(60∘N,25∘E)R(60^\circ N, 25^\circ E) along the parallel of latitude 60∘N60^\circ N.
    The distance along a parallel of latitude is given by D=REcos⁡LΔλD = R_E \cos L \Delta \lambda, where Δλ\Delta \lambda is the difference in longitude in radians.
    Here, L=60∘L = 60^\circ.
    λP=35∘W\lambda_P = 35^\circ W, λR=25∘E\lambda_R = 25^\circ E.
    The difference in longitude Δλ=35∘+25∘=60∘\Delta \lambda = 35^\circ + 25^\circ = 60^\circ.
    Convert Δλ\Delta \lambda to radians: Δλ=60∘×π180∘=π3\Delta \lambda = 60^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{3} radians.

    DB=6370 km×cos⁡60∘×π3D_B = 6370 \, \text{km} \times \cos 60^\circ \times \frac{\pi}{3} DB=6370 km×12×π3D_B = 6370 \, \text{km} \times \frac{1}{2} \times \frac{\pi}{3} DB=6370π6≈3337.8 kmD_B = \frac{6370 \pi}{6} \approx 3337.8 \, \text{km}

    Time taken by Airliner B: TB=DBSpeedT_B = \frac{D_B}{\text{Speed}}.

    TB=3337.8 km600 km/hr≈5.563 hoursT_B = \frac{3337.8 \, \text{km}}{600 \, \text{km/hr}} \approx 5.563 \, \text{hours}
  3. Compare arrival times.

    TA≈8.338 hoursT_A \approx 8.338 \, \text{hours}
    TB≈5.563 hoursT_B \approx 5.563 \, \text{hours}
    Airliner B arrives earlier.

  4. Calculate the difference in arrival time in minutes.

    Difference in time =TA−TB=8.338−5.563=2.775 hours= T_A - T_B = 8.338 - 5.563 = 2.775 \, \text{hours}.
    Convert to minutes: 2.775 hours×60 minutes/hour=166.5 minutes2.775 \, \text{hours} \times 60 \, \text{minutes/hour} = 166.5 \, \text{minutes}.
    Rounding to the nearest minute, the difference is 167 minutes167 \, \text{minutes}.

    Airliner B arrives earlier by approximately 167 minutes.

    The final answer is Airliner B arrives earlier by 167 minutes\text{Airliner B arrives earlier by 167 minutes}.

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