A taxi driver's satnav map is drawn to scale, so the triangular route between three towns, X, Y and Z, on the map is similar to the actual route triangle. X...
Question 1 Report
A taxi driver's satnav map is drawn to scale, so the triangular route between three towns, X, Y and Z, on the map is similar to the actual route triangle.
On the map, \(XY = 4.5\) cm, \(YZ = 6\) cm and \(XZ = 7.5\) cm. In real life the actual distance \(XY\) is \(27\) km.
- Find the scale factor of the map, map to real life, giving your answer as a ratio in the form \(1 : n\). (2)
- Work out the real-life distances \(YZ\) and \(XZ\). (2)
- Taxi fares on this route are charged at £1.20 per km. Work out the total cost of the journey from X to Y to Z and back to X. (2)
Answer Details
A map scale expresses the ratio of a distance on the map to the corresponding real-life distance, and once found, the same ratio converts every other length on the map to real life.
(a) Converting the real distance to centimetres to match the map's units: \(27\) km \(=27\times1000\times100=2700000\) cm. The scale factor, map to real life, is
\[XY_{\text{map}}:XY_{\text{real}}=4.5:2700000\]Dividing both sides by 4.5 to get the ratio into the form \(1:n\):
\[1:600000\] [2 marks](b) Multiplying each map length by the scale factor 600000 (then converting back to km):
\[YZ=6\times600000=3600000 \text{ cm}=36 \text{ km}\] \[XZ=7.5\times600000=4500000 \text{ cm}=45 \text{ km}\] [2 marks](c) The full round trip covers \(XY+YZ+ZX=27+36+45=108\) km. At £1.20 per km:
\[108\times1.20=\pounds129.60\] [2 marks]Converting the real-life distance into the map's units (cm) before dividing is essential; comparing 4.5 cm directly with 27 km without a unit conversion would give a scale factor with the wrong number of zeros.
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