A bus company is considering a new direct road between two towns. Currently, buses travel via junction \(B\), with \(AB = 14\) km, \(BC = 20\) km and angle ...
Question 1 Report
A bus company is considering a new direct road between two towns. Currently, buses travel via junction \(B\), with \(AB = 14\) km, \(BC = 20\) km and angle \(B = 75^\circ\), shown below.
- Use the cosine rule to calculate the length \(AC\) of the proposed direct road, correct to 3 significant figures. (2)
- Work out the length of the current route via \(B\). (1)
- Determine, showing your working, how much shorter the direct road would be than the current route. (2)
Answer Details
The cosine rule finds the direct distance \(AC\) using the two known legs of the current route and the angle between them at \(B\); the saving is then the difference between the two-leg route length and this direct distance.
- \(AC^2=14^2+20^2-2\times14\times20\times\cos75^\circ\) [1 mark]; \(AC=21.2\) km (3 s.f.). [1 mark]
- Current route \(=AB+BC=14+20=34\) km. [1 mark]
- Difference \(=34-21.2=12.8\) km. [1 mark] So the direct road would be \(12.8\) km shorter than the current route via \(B\). [1 mark]
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