Clock Angle Formula Application

To find the angle between the hour and minute hands of a clock, we use the formula:

Angle = $ |(30 \times H - \frac{11}{2} \times M)| $

Where H is the hour and M is the minute.

Calculating Angle at 5:24

Given time is 5:24. So, H = 5 and M = 24.

Substitute the values into the formula:

Angle = $ |(30 \times 5 - \frac{11}{2} \times 24)| $

First, calculate the position of the minute hand:

Minute hand position = $ \frac{11}{2} \times 24 = 11 \times 12 = 132 $ degrees from the 12.

Next, calculate the position of the hour hand:

Hour hand position = $ 30 \times 5 = 150 $ degrees from the 12.

Now, find the difference:

Angle = $ |150 - 132| $

Angle = $ |18| $

Angle = $ 18^\circ $

Determining the Smaller Angle

The calculated angle is $ 18^\circ $. Since this value is less than $ 180^\circ $, it represents the smaller angle between the hands.