IMC 2011 Q19
19. Harrogate is 23km due north of Leeds, York is 30km due east of Harrogate, Doncaster is 48km due south of York, and Manchester is 70km due west of Doncaster. To the nearest kilometre, how far is it from Leeds to Manchester, as the crow flies?
- A 38km
- B 47km
- C 56km
- D 65km
- E 74km
Intermediate Mathematical Challenge 2011, Question 19. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: B 47km
As can be seen from the diagram, Manchester is 40km west of Leeds and 25 km south of it.
Therefore, by Pythagoras' Theorem, the distance in km from Leeds to Manchester as the crow flies is
\(\sqrt{25^2 + 40^2} = 5\sqrt{5^2 + 8^2} = 5\sqrt{89}\).
Now \(\sqrt{81} = 9\) and \(\sqrt{100} = 10\), so \(9 < \sqrt{89} < 10\).
This means that the required distance is between 45km and 50km and, of the options given, only 47km (corresponding to the approximation \(\sqrt{89} \approx 9.4\)) lies in this interval.
(Please note that the distances given in this problem are all approximate.)
Solution © UK Mathematics Trust, from the IMC 2011 solutions. Longer versions with investigations: Solutions and investigations (PDF).