There are 60 minute graduations around the 360 degree clock face, so the minute
graduations are 360/60 = 6 degrees apart. Therefore the minute hand's angular
speed is 6 degrees per minute or 6 dpm.
The minute hand rotates 12 times as fast as the hour hand, so the hour hand's
angular speed is 1/12 as fast, or 6/12 = 1/2 = 0.5 dpm.
So the angle between the hands is increasing at 6.0-0.5 = 5.5 dpm. Every time
the hands are exactly together they are 0 degrees or 360 degrees apart.
Notice that we can always add (or subtract) 360 degrees to (or from) any angle
between the hands or between the vertical and either hand without incurring any
difficulty.
We will also use the formulas:
Degrees rotated = (degrees per minute)(number of minutes)
and
Number of minutes = (degrees rotated)/(degrees per minute)
It is now between 9 and 10 o’clock.
a.)At what time after 9’oclock will the minute hand and the hour hand be
perpendicular for the first time?
At 9 o'clock, the angle between the hands is 90 degrees and the first time they
will be perpendicular again is when the angle between them is 270 degrees.
So the angle between them must increase by 270-90=180 degrees at the rate of
5.5 dpm which will be 180/5.5=1800/55=360/11 = 32 8/11.
Answer: The time will be 9:32 8/11.
b.)In 4 minutes, the hour hand of the clock will be directly opposite the
position occupied by the minute hand 3 minutes ago. What time is it?
Let that time (which we consider as 'now') be x minutes after 9 o'clock.
At 9 o'clock, the minute hand was straight up, or 0 or 360 degrees with the
vertical. So now, x minutes later, the minute hand, which has been rotating at
6 dpm, is now at an angle of 6x degrees with the vertical, which, for
convenience, we can consider as 6x+360 degrees with the vertical.
At 9 o'clock, the hour hand was "directly left" or 270 degrees with the
vertical. So now, x minutes later, the hour hand, which has been rotating at
0.5 dpm, is now at an angle of 270+0.5x degrees with the vertical.
So right now, x minutes after 9 o'clock, the minute hand makes an angle of
6x+360 degrees with the vertical and the hour hand makes an angle of 270+0.5x
degrees with the vertical.
4 minutes from now, the hour hand, which will be rotating at 0.5 dpm, will have
rotated through (0.5)(4)=2 more degrees, which will put it at the position of
270+0.5x+3 or 273+0.5x degrees with the vertical, 4 minutes from now.
3 minutes ago, the minute hand, which has been rotating at 6 dpm, has now
rotated through (6)(3)=18 more degrees since 3 minutes ago. That means that 3
minutes ago, it was at 6x+360-18 or 6x+342 degrees with the horizontal.
The position directly opposite 6x+342 is found by subtracting 180 degrees from
it, which is 6x+342-180, or 6x+162.
Now the equation is
273+0.5x = 6x+162
111 = 5.5x
111/5.5 = x
111/5.5 = 1110/55 = 222/11 = 20 2/11 minutes after 9 o'clock.
Answer: The time is 9:20 2/11
c.)In a quarter of an hour the minute hand will be behind the hour hand by only
half as much as is now behind it. What time is it?
Let that time (which we consider as 'now') be x minutes after 9 o'clock.
As explained in part b), the minute hand now makes an angle of 6x, or 6x+360
degrees with the vertical, and the hour hand now makes an angle of 270+0.5x
degrees with the vertical.
So the number of degrees the minute hand is now behind the hour hand is their
difference (270+0.5x)-(6x+360)=270+0.5x-6x-360=-5.5x-90 degrees and we can
always add 360 degrees, so we can consider the number of degrees the minute hand
is now behind the hour hand as -5.5x-90+360 or 270-5.5x.
In a quarter of an hour (15 minutes), the minute hand, rotating at 6 dpm, will
have rotated through (6)(15)=90 degrees. So the position of the minute hand,
rotating at 6 dpm, will then be 6x+360+90 or 6x+450 degrees.
In a quarter of an hour (15 minutes), the hour hand, rotating at 0.5 dpm, will
have rotated through (0.5)(15)=7.5 degrees. So the position of the hour hand,
rotating at 6 dpm, will then be 270+0.5x+7.5 or 277.5+0.5x degrees with the
vertical.
The number of degrees the minute hand will then be behind the hour hand is
their difference, or (277.5+0.5x)-(6x+450) or 277.5+0.5x-6x-450=-172.5-5.5x, and
to keep that from being negative we can add 360 degrees, making it
-172.5-5.5x+360 or 187.5-5.5x.
So the equation is
187.5-5.5x = (1/2)(270-5.5x)
Multiply through by 2
375-11x = 270-5.5x
105 = 5.5x
105/5.5 = x
1050/55 = x
210/11 = x
19 1/11 = x
Answer: The time is 9:19 1/11
Edwin
