A drone company compares two routes from its hub \(H\). Route A flies \(6\) km to \(S_1\), then \(9\) km to \(S_2\), turning through \(78^{\circ}\) at \(S_1...
Question 1 Report
A drone company compares two routes from its hub \(H\). Route A flies \(6\) km to \(S_1\), then \(9\) km to \(S_2\), turning through \(78^{\circ}\) at \(S_1\). Route B flies \(7\) km to \(S_3\), then \(8\) km to \(S_4\), turning through \(95^{\circ}\) at \(S_3\), as shown.
- Use the cosine rule to find the direct distance \(HS_2\), correct to 3 significant figures. (2)
- Use the cosine rule to find the direct distance \(HS_4\), correct to 3 significant figures. (2)
- At the same flying speed, state, with a reason, which route is quicker. (2)
Answer Details
Each route forms a triangle with two known side lengths and the angle turned through at the middle point; the cosine rule, \(a^{2}=b^{2}+c^{2}-2bc\cos A\), converts each turning angle and pair of leg lengths into the direct "as the crow flies" distance, and comparing the two direct distances at a fixed speed decides which route is quicker.
(a) Applying the cosine rule to Route A's triangle: \(HS_2^{2}=6^{2}+9^{2}-2(6)(9)\cos78^{\circ}=36+81-108\cos78^{\circ}\approx94.5\), so \(HS_2\approx9.72\) km (3 s.f.). [2 marks]
(b) Applying the cosine rule to Route B's triangle: \(HS_4^{2}=7^{2}+8^{2}-2(7)(8)\cos95^{\circ}=49+64-112\cos95^{\circ}\approx122.8\), so \(HS_4\approx11.1\) km (3 s.f.). [2 marks]
(c) At the same flying speed, time is proportional to distance, so the route with the shorter direct distance is quicker: since \(9.72\lt11.1\), Route A is quicker. [2 marks]
The turning angle used in each cosine rule calculation is the angle actually turned through at the intermediate stop, not its supplement; using the wrong one of these two related angles is a common source of error whenever a journey's direct distance is found from two legs and a turn.
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