This investigation explores using different shapes as the hands of the clock. What things occur as the the hands move.
During the third hour after midnight the hands on a clock point in the same direction (so one hand is over the top of the other). At what time, to the nearest second, does this happen?
How many times in twelve hours do the hands of a clock form a right angle? Use the interactivity to check your answers.
Well done to those of you who sent in the correct answer to this problem. A lot of you explained your reasoning very carefully, but sadly we are unable to name you all here.
In $10$ minutes, the minute hand will sweep $60$ $^\circ$ degrees, because in $5$ minutes the minute hand will sweep $360\div12$ = $30$ $^\circ$.
In $3$ hours, the hour hand will sweep $90$ $^\circ$ degrees, because in $1$ hour the hour hand will sweep $360\div12$ = $30$ $^\circ$ .
If the minute hand goes through $180$ $^\circ$, the hour hand will sweep $15$ $^\circ$. The reason is as follows:
When the minute hand goes through $180$ $^\circ$, it is half an hour. In $1$ hour the hour hand will sweep $360\div12$ = $30$ $^\circ$ and $30$ $^\circ$ $\div2$ = $15$ $^\circ$. So when the minute hand sweeps $180$ $^\circ$, the hour hand will turn $15$ $^\circ$.
Joshua from Sydney Grammar School had a slightly different approach to the second part of the problem:
In three hours, the hour hand will travel a quarter of a full revolution, which is $90$ $^\circ$.
For those of you who misread this question and calculated the number of degrees that the minute hand turned through, better luck next time!