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| {{unreferenced|date=October 2013}}
| | Wilber Berryhill is what his wife loves to call him and he completely loves this name. To climb is some thing I truly appreciate performing. Her family members lives in Ohio but her husband desires them to transfer. Office supervising is where her primary income arrives from.<br><br>My blog - clairvoyants ([http://alles-herunterladen.de/excellent-advice-for-picking-the-ideal-hobby/ alles-herunterladen.de]) |
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| [[Image:clockAngles.jpg|thumb|The diagram shows the angles formed by the hands of an analog clock showing a time of 2:20]]
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| '''Clock angle problems''' are a type of [[mathematical problem]] which involve finding the angles between the hands of an [[analog clock]].
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| ==Math problem==
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| Clock angle problems relate two different measurements: [[angle]]s and [[time]]. The angle is typically measured in [[degree (angle)|degrees]] from the mark of number 12 clockwise. The time is usually based on [[12-hour clock]].
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| A method to solve such problems is to consider the rate of change of the angle in degrees per minute. The hour hand of a normal 12-hour analogue clock turns 360° in 12 hours (720 minutes) or 0.5° per minute. The minute hand rotates through 360° in 60 minutes or 6° per minute.
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| ===Equation for the angle of the hour hand===
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| <math>\theta_{\text{hr}} = \frac{1}{2}M_\Sigma = \frac{1}{2}(60H + M)</math>
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| where: | |
| * <math>\scriptstyle\theta</math> is the angle in degrees of the hand measured clockwise from the 12 o'clock position.
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| * <math>\scriptstyle H</math> is the hours past 12 o'clock.
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| * <math>\scriptstyle M</math> is the minutes past the hour.
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| * <math>\scriptstyle M_\Sigma</math> is the minutes past 12 o'clock.
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| ===Equation for the angle of the minute hand===
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| <math>\theta_{\text{min.}} = 6M</math>
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| where:
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| * <math>\scriptstyle\theta</math> is the angle in degrees of the hand measured clockwise from the 12 o'clock position.
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| * <math>\scriptstyle M</math> is the minute.
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| ====Example====
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| The time is 5:24. The angle in degrees of the hour hand is:
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| <math>\theta_{\text{hr}} = \frac{1}{2}(60 \times 5 + 24) = 162</math>
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| The angle in degrees of the minute hand is:
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| <math>\theta_{\text{min.}} = 6 \times 24 = 144</math>
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| ===Equation for the angle between the hands===
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| The angle between the hands can be found using the formula:
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| <math>\begin{align}
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| \Delta\theta
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| &= \left|\theta_{\text{hr}} - \theta_{\text{min.}}\right| \\
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| &= \left|\frac{1}{2}(60H + M) - 6M\right|\\
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| &= \left|\frac{1}{2}(60H - 11M)\right|
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| \end{align}</math>
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| where
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| * <math>\scriptstyle H</math> is the hour
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| * <math>\scriptstyle M</math> is the minute
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| ====Example====
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| The time is 2:20.
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| <math>\begin{align}
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| \Delta\theta
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| &= \left|\frac{1}{2}(60 \times 2 - 11 \times 20)\right|\\
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| &= \left|\frac{1}{2}(120 - 220)\right|\\
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| &= 50
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| \end{align}</math>
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| ===When are the hour and minute hands of a clock superimposed?===
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| The hour and minute hands are superimposed only when their angle is the same.
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| <math>\begin{align}
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| \theta_{\text{hr}} &= \theta_{\text{min.}}\\
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| \Rightarrow \frac{1}{2}(60H + M) &= 6M\\
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| \Rightarrow 11M &= 60H\\
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| \Rightarrow M &= \frac{60}{11}H\\
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| \Rightarrow M &= 5.\overline{45}H
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| \end{align}</math>
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| <math>\scriptstyle H</math> is an integer in the range 0–11. This gives times of: 0:00, 1:05.{{overline|45}}, 2:10.{{overline|90}}, 3:16.{{overline|36}}, etc. (0.{{overline|45}} minutes are exactly 27.{{overline|27}} seconds.)
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| ==See also==
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| *[[Clock position]]
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| ==External links==
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| * http://www.delphiforfun.org/Programs/clock_angle.htm
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| * http://www.ldlewis.com/hospital_clock/ - extensive clock angle analysis
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| * http://www.jimloy.com/puzz/clock1.htm
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| [[Category:Mathematics education]]
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| [[Category:Elementary mathematics]]
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| [[Category:Elementary geometry]]
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| [[Category:Mathematical problems]]
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| [[Category:Clocks]]
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Wilber Berryhill is what his wife loves to call him and he completely loves this name. To climb is some thing I truly appreciate performing. Her family members lives in Ohio but her husband desires them to transfer. Office supervising is where her primary income arrives from.
My blog - clairvoyants (alles-herunterladen.de)