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At what time between $6$ a. m. and $7$ a. m. will the minute hand and hour hand of a clock make an angle closest to $60°$?

  1. $ 6: 22$ a.m.
  2. $6: 27$ a.m.
  3. $ 6: 38$ a.m.
  4. $ 6: 45$ a.m.

My attempt :

The angle between the hands can be found using the following formula:

\begin{align} \Delta\theta &= \vert \theta_{\text{hr}} - \theta_{\text{min.}} \vert \\ &= \vert 0.5^{\circ}\times(60\times H+M) -6^{\circ}\times M \vert \\ &= \vert 0.5^{\circ}\times(60\times H+M) -0.5^{\circ}\times 12 \times M \vert \\ &= \vert 0.5^{\circ}\times(60\times H -11 \times M) \vert \\ \end{align} where

H is the hour M is the minute If the angle is greater than 180 degrees then subtract it from 360 degrees.

Therefore :

  1. $59^{\circ}$
  2. $31.5^{\circ}$
  3. $29^{\circ}$
  4. $67.5^{\circ}$

Are my approach and solution correct ? Any other way to solve this ?

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HINT : The hour hand moves $1/2$ degrees per minute while minute hand moves 6 degrees per minute. Hope this helps you.Thus it would be near to 60 at 6:22.so now here at 22 past 6 the hour hand would go 11 degrees while the minute hand would go 22.6=132 degrees so as per you stated it would be 180-132=48 so total angle would be 48+11 =59 nearest to 60.

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  • $\begingroup$ Sorry for the typo. $\endgroup$ – Archis Welankar Nov 10 '15 at 12:44

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