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# Help

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What is the measure, in degrees, of the acute angle formed by the hour hand and the minute hand of a 12-hour clock at 6:48?

Jan 29, 2019

#1
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$$\text{let's use the navigational assignment of degrees to the clock, i.e. }\\ \text{12 is 0 degrees increasing as we move clockwise back to 360 degrees again at 12}\\ \phi_m = 360\dfrac{min}{60} = 360\dfrac{48}{60} = 360 \dfrac 4 5 = 72\cdot 4 = 288^\circ\\ \phi_h = 360 \dfrac{60hr + min}{720} = 360 \dfrac{60\cdot 6 + 48}{720} = \dfrac 1 2(360+48) = 204^\circ\\ |288^\circ - 204^\circ| = 84^\circ$$

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Jan 29, 2019
#2
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You can use the clock angle formula, which is $$|0.5\times (60\times H-11\times M)|$$, where H is the number of hours and M is the number of minutes.

$$|0.5\times (60\times 6-11\times 48)| = |0.5\times (360-528)| = |0.5\times (-168)|=|-84|=84$$

Therefore, the angle between the hands is 84 degrees.

You are very welcome!

:P

Jan 29, 2019
#3
+3

The shortened version of the formula is: $$|30H-5.5M|$$, where $$H$$ stands for hours and $$M$$ stands for minutes. Thus, we have $$|30(6)-5.5(48)|=|180-264|=\boxed{84}$$ degrees.

Jan 29, 2019
#4
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Thanks tertre.....I'm going to have to remember that formula!!!!   CPhill  Jan 29, 2019
#5
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Yes, it's pretty neat!

tertre  Jan 30, 2019