An aeroplane flies from a town P(lat. 40^\circN, 38^\circE) to another town Q(lat. 40^\circN, 22^\circW). It later flies to a third town T(28^\circN, 22^\circW). Calculate the :
(a) distance between P and Q along their parallel of latitude ;
(b) distance between Q and T along their line of longitudes;
(c) average speed at which the aeroplane will fly from P to T via Q, if the journey takes 12 hours, correct to 3 significant figures. [Take the radius of the earth = 6400km ; \(\pi = 3.142\)]
Verified Correct Answer: .
Step-by-Step Mathematical Solution
(a) Distance along arc PQ = \(\frac{\theta}{360^\circ} \times 2 \pi r\)
where \(r = R \cos \theta\)
\(|PQ| = \frac{60^\circ}{360^\circ} \times 2 \times \frac{22}{7} \times 6400 \times \cos 40^\circ\)
\(|PQ| = \frac{1}{6} \times 2 \times \frac{22}{7} \times 6400 \times 0.7660\)
= \(5134.74 km \approxeq 5135km\) (to the nearest whole number)
(b) Difference in latitude between Q and T = 40^\circ - 28^\circ = 12^\circ
\(\therefore |QT| \text{along the line of longitude} = \frac{12}{360} \times 2 \times \frac{22}{7} \times 6400\)
\(\frac{1}{30} \times 2 \times \frac{22}{7} \times 6400 = 1340.95km \approxeq 1341km\)
(c) Average speed = \(\frac{\text{Total distance covered}}{\text{Total time taken}}\)
= \(\frac{5134.74 + 1340.95}{12} = 539.64km/hr\)
\(\approxeq 540km/hr\) (to 3 significant figures)
Source: Myschool Past Questions Archive (Record #54652)