Question

**HELP**
To the nearest second, what is the 1st time after 7:00 that the hand of a clock form at right angle?

Format answer as D:M:S

Answer 1

The minute hand sweeps out 360 deg in 3600 s (1 hour)

It starts at 0 deg (12 on the face)

So its angle at any time t is

anglem = 0+360t/3600 = t/10

The hour hand sweeps out 30 deg in 3600 s (1 hour )

It starts at 210 deg (7 on the clock face)

Therefore its angle at any time t is

angleh = 210 +30t/3600 = 210 +t/120

You need to find when angleh = anglem+90 (i.e. a right angle)

so
210+t/120 = t/10 +90

Solve this for t to get the answer in seconds.

You then need to convert the seconds to get the minutes and seconds after 7 oclock

(Hint - the answer is between 20 and 25 minutes)

Answer 2

Algebra is an excellent tool to solve problems.

There are times in our daily lives that we do not have a pen or paper, or a young child has simply not yet learned algebra, that mathematical reasoning can be another powerful tool for problems solving.

We note that the hours hand moves slower than the minutes hand, to the extent of 1/12.
So the hours hand is actually moving 11/12 minute ahead of the hours hand for every minute it moves.

For example: \(to\ the\ nearest\ second,\ when\ do\ the\ two\ hands\ overlap\ since\ 6\ o'clock?\)
We know that if the hours hand does not move, the minutes hand must move 30 minutes.
If it does, it will have to move (12/11)*30=360/11 minutes, or 32 8/11 minutes, or 32 minutes 43.64 seconds approximately.

Answer 3

*So the \(minutes\) hand is actually moving 11/12 minute ahead of the hours hand for every minute it moves.