4 Answers By Expert Tutors
It helps to create expressions for the angular position of the minute hand and the hour hand, then to set them 90 degrees apart. Hour hand: 360 degrees with 12 hours per circle means 360/12=30 degrees per hour or 0.5 degrees per minute Minute hand: 360 degrees with 60 minutes per circle means 360/60=6 degrees per minuteLet t = time (elapsed since 5:00 in minutes with decimals) H = angle of hour hand (0-360 degrees clockwise) M = angle of minute hand (0-360 degrees clockwise) H = 150 + 0.5t [angle for 5 o’clock plus minutes] M = 0 + 6t [angle for minutes] “perpendicular” means 90 degrees apart (H - M) = 90 or - (H - M) = 90 (1) (H - M) = 90 (150 + 0.5t) - (0 + 6t) = 90 t = 10.909 minutes after 5, or 5:10:55 (hh:mm:ss) or (2) -(H - M) = 90 (0 + 6t) - (150 + 0.5t) = 90 t = 43.636 minutes after 5, or 5:43:38 (hh:mm:ss)
John M. answered • 12/24/16
Engineering manager professional, proficient in all levels of Math
- The hour hand has to travel 360 degrees in 12 hours. For example, if its 1 AM, the hour hand is on the 1. 12 hours later, it is 1PM and the hour hand is back on the 1. So, 360 degrees in 1 hour.
- HOUR HAND: 360 deg/ 12* 60 min = 360/ (720) = 0.5 deg/min
- The minute hand has to travel 360 degrees in 1 hour
- MINUTE HAND: 360 deg/ 60 min = 6 deg/min
- In order to be perpendicular, the two hands need to form a 90 degree angle.
- Start at 5:00. At this time, the Hour hand is on the 5 and the minute hand are aligned at 12. That is, they are separated by 150 deg.
- For each minute that goes by, the minute hand travels 6 deg and the hour hand at 0.5 deg.
- So, now we can develop the equation, where x is the minutes after 5 o'clock:
- 6x + 90 = 0.5x + 150
- 5.5x = 60
- x = 60/5.5
- x = 10.91
- So, at 5:00 + 10.91 minutes, the hour and minute hand will be perpendicular. This equates to roughly 5:10:55
- But now there is another opportunity for the minute hand and hour hand to be perpendicular. It will happen again as the minute hand passes the hour hand. And the equation for this is:
- 6x - 90 = 0.5x + 150
- 5.5x = 240
- x = 43.63 min
- So, at 5:00 + 43.63 minutes, the hour and minute hand will again be perpendicular. This equates to roughly 5:43:38
At 5:00 the angle is 150°.
Every minute the minute hand moves the angle is decreased by 6°
So for every minute the angle decreases, 150 - 6m.
Yet the hour hand is moving forward also.
It moves one tick, 6...°, for every 12 minutes of the minute hand.
So the angle moves at 150 - 6m + 6m/12
For what value of m does that equal 90?
At 5:30 the hands form a 15 degree angle.
Min hand moves 6 degrees every min
Hour hand moves 1/2 degree every min
Therefore, the degree between them changes 5.5 degrees every min.
Around 43 and 7/11 minutes after 5
Between 5:43 and 5:44
Option 3 : \(7:21\frac{9}{{11}}\)
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Angle made by the hour hand with the minute hand at a given time is \(\frac{{60H - 11M}}{2}\)
Where H is the hour and M is the minute.
We have H = 7 and M is unknown. Also, the angle should be \(90^\circ \).
Therefore, \(90 = \frac{{60 \times 7 - 11M}}{2} \Rightarrow 180 = 420 - 11M \Rightarrow 11M = 420 - 180 = 240\)
Thus, we get \(M = \frac{{240}}{{11}} = 21\frac{9}{{11}}\)
Hence, the correct answer is \(7:21\frac{9}{{11}}\).
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Oh, let's do this the annoying way.
The minute hand moves at $360\frac {\text{degrees}}{\text{revolutions}}\frac {1 \text{revolution}}{60 \text{minutes}}=6\frac{\text{degrees}}{\text{minutes}}$
The hour hand moves at $\frac{360 degrees}{1revolution}\frac{1revolution}{12hours}\frac{1hour}{60 minute}= \frac 12 \frac{degree}{minute}$.
If a minute hand is $x$ degrees away from the hour hand how long will it take for the minute hand to become $x + 180$ degrees away from the hour hand?
Well that's a matter of solving $x + 6t = x + \frac 12t + 180$ or $t= \frac {180}{\frac {11}{2}}=\frac {360}{11}$. (Roughly $32$ minutes.)
At $3:00$ the hour hand and the minute hand are perpendicular. The next time that will happen will be when the minute hand moves for $90$ degrees before the hour had to $90$ degrees after the hour. That will happen in about $32$ minutes. (At $3:32$). Every $\frac {360}{11}$ theminute hand will move 180 degrees further than the hour hand and the hands will be perpendicular.
In a $12$ hour (or a $60*12 = 720$ minute period) this will happen $\frac {720}{\frac {360}{11}} = 22$ times.
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Here's a better way, although it was easy to make annoying sign and logic errors if you aren't careful.
The minute hand travels $6$ degrees a minute. ($\frac {360}{60} = 6$). The hour hand travels $.5$ degrees a minute. ($\frac {360}{12*60} = .5$).
In a 12-hour period there are $12*60=720$ minutes.
Let's say $\theta_t = 6t$ is the angle of the minute hand after $t; 0 \le t < 720$ minutes. Note: it's very possible that $\theta(t) > 360$.
Let's say $\phi_t = .5t$ is the angle of the hour hand after $t$ minutes..
Obviously $\theta_t \ge \phi_t$.
$\theta_t$ and $\phi_t$ are perpendicular if $\theta_t = \phi_t + 90 + k*180$ for some non-negative integer $k$.
So we need the find out how many solutions there are to:
$6t = .5t + 90 + k*180; 0 \le t < 720$ there are.
So $t = \frac {90 + k*180}{5.5}$ will have one solution for each integer $k$.
So how many $k$ are there so that $0 \le t = \frac {90 + k*180}{5.5} < 720$ are there?
Multiply everything by $11$: $0 \le 2(90 + k*180)< 11*720$
Divide everything by $180$: $0 \le 1 + 2k < 11*4$
$-1 \le 2k < 11*4 -1$
$\frac 12 \le k < 2*11 - \frac 12$
And as $k$ is a non-negative integer $0 \le k < 22$. There are $22$ possible $k$s which give a solution.
This also gives us the times this occurs:
$t = \frac {90 + k*180}{5.5} = \frac {180}11 + k *\frac {360}{11} \approx 16.36 + 32.72k$ (or a little more than every half-hour):
So at 12:16, 12:49, 1:22, 1:54, 2:27, 3:00, etc.