A taxi company compares two routes between the same start and end point. Route A is a direct journey represented by the vector \(\mathbf{a} = \binom{8}{16}\...
Question 1 Report
A taxi company compares two routes between the same start and end point. Route A is a direct journey represented by the vector \(\mathbf{a} = \binom{8}{16}\), measured in kilometres. Route B consists of two stages, \(\binom{3}{4}\) followed by \(\binom{5}{12}\).
- Find \(|\mathbf{a}|\), the distance for Route A, giving your answer as a simplified surd. (2)
- Find the total distance travelled along Route B. (2)
- Given that \(8\sqrt5 \approx 17.9\), state which route is shorter, giving a reason. (1)
Answer Details
Route A's length is the magnitude of a single vector, found using Pythagoras' theorem, while Route B's length is the sum of the magnitudes of its two separate stages; comparing the two totals then decides which route is shorter.
(a) \(|\mathbf{a}|=\sqrt{8^{2}+16^{2}}=\sqrt{64+256}=\sqrt{320}\). Since \(320=64\times5\), this simplifies to \(\sqrt{64}\times\sqrt5=8\sqrt5\) km. [2 marks]
(b) The first stage has length \(\sqrt{3^{2}+4^{2}}=\sqrt{25}=5\) km; the second stage has length \(\sqrt{5^{2}+12^{2}}=\sqrt{169}=13\) km. The total distance for Route B is \(5+13=18\) km. [2 marks]
(c) Since \(8\sqrt5\approx17.9\) is less than \(18\), Route A, the direct journey, is the shorter route. [1 mark]
Route B's stages, \(\binom34\) and \(\binom{5}{12}\), are both scaled versions of the well-known \(3\)-\(4\)-\(5\) Pythagorean triple, which is why their magnitudes come out as the exact whole numbers \(5\) and \(13\) rather than surds.
Download The App On Google Playstore
Everything you need to excel in your exams
