Question

What is the angle between the hands of the clock at $5:30$ Pm?

Answer 1

Hint: We need to solve this question by using the concept of angles. We need to see the angle by which the hour hand and minute hands move and then compare the values for the given time in order to calculate the angle between the two hands at the specified time.

Complete step by step answer:
In order to solve this question, let us use the concept of a circle having $360{}^\circ .$ The clock is a circle and therefore, it had $360{}^\circ .$ For a full circle of the hour hand, we cover a total of 12 hours. This means that in one hour, it will cover an angle of
$\Rightarrow \dfrac{360{}^\circ }{12}=30{}^\circ $
It is found that the hour hand moves a total of $30{}^\circ $ in one hour or 60 minutes. Now, for 1 minute, we can say that the hour hand moves by
$\Rightarrow \dfrac{30{}^\circ }{60}=0.5{}^\circ $
For the given time, we have represented the time on the clock as shown below.

The given time is $5:30$ Pm which means that the total angle covered by the hour hand is given by the movement for 5 hours and 30 minutes.
$\Rightarrow 30{}^\circ \times 5+0.5{}^\circ \times 30=150{}^\circ +15{}^\circ $
Adding the above terms,
$\Rightarrow 165{}^\circ $
Therefore, we can say that the hour hand moves by a total of $165{}^\circ $ for the given time.
Similarly, for the minute hand, we can say that in 60 minutes it covers 360 degrees. This means that for 1 minute it covers,
$\Rightarrow \dfrac{360{}^\circ }{60}=6{}^\circ $
Therefore, for the given 30 minutes it starts from $0{}^\circ $ and covers a distance of
$\Rightarrow 6{}^\circ \times 30=180{}^\circ $
Now the difference between the minute hand angle and hour hand angle is given by,
$\Rightarrow 180{}^\circ -165{}^\circ =15{}^\circ $
Hence, the angle between the hands of the clock at $5:30$ Pm is $15{}^\circ .$

Note: We need to note that the movement of the hour hand due to the minutes passed should also be considered. Without this, we could get wrong answers. This account is due to the hour hand moving gradually throughout the 1-hour time. It does not suddenly jump to the next hour.