Robo-Turn
Weekly Problem 16 - 2007
Can you figure out how far the robot has travelled by the time it is first facing due East?
Can you figure out how far the robot has travelled by the time it is first facing due East?
Age
11 to 14
Challenge level
A robot, which is initially facing North, is programmed to travel 5m then turn through $10^{\circ}$, travel 5m then turn through $20^{\circ}$ clockwise, travel 5m then turn through $30^{\circ}$ clockwise, and so on. Each move consists of moving 5m in a straight line and then turning clockwise through an angle which increases by $10^{\circ}$ clockwise at each move.
How far has it travelled by the time it is first facing due East at the end of a move?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
45m
The total angle turned through after each of the first 4 moves is $10^{\circ}$, $30^{\circ}$, $60^{\circ}$ and $100^{\circ}$. So the robot does not face due East at the end of a move in its first complete revolution. The total angle it has turned through after each of the next 5 moves is $150^{\circ}$, $210^{\circ}$, $280^{\circ}$, $360^{\circ}$ and $450^{\circ}$, so at the end of the 9th move the robot does face due East. As the robot moves 5m in each move, the distance it travels is 45m.
Note that this solution assumes that the robot is not starting the process close to the North Pole!