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 $60°$?
- $ 6: 22$ a.m.
- $6: 27$ a.m.
- $ 6: 38$ a.m.
- $ 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 :
- $59^{\circ}$
- $31.5^{\circ}$
- $29^{\circ}$
- $67.5^{\circ}$
Are my approach and solution correct ? Any other way to solve this ?
Best Answer
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.