Calculating the Angle Between Clock Hands at 4:52 p.m.

This problem involves calculating the angle between the hour hand and the minute hand of a standard 12-hour analog clock at a specific time, 4:52 p.m. To solve this, we need to determine the position (angle from the 12 mark) of both hands at that exact moment.

Understanding Clock Hand Movement

• A clock face is a circle, covering 360 degrees.
• There are 12 hours marked on the clock face.
• The space between two consecutive hour marks represents $360^\circ / 12 = 30^\circ$.

Let's analyze the movement of each hand:

• Minute Hand: The minute hand completes a full circle (360 degrees) in 60 minutes. This means its speed is $360^\circ / 60 \text{ minutes} = 6^\circ$ per minute.
• Hour Hand: The hour hand completes a full circle (360 degrees) in 12 hours. This means its speed is $360^\circ / 12 \text{ hours} = 30^\circ$ per hour. In terms of minutes, its speed is $30^\circ / 60 \text{ minutes} = 0.5^\circ$ per minute.

Position of Hands at 4:52 p.m.

We will calculate the angle of each hand relative to the 12 o'clock mark, moving clockwise.

Minute Hand Position:

At 4:52 p.m., the minute hand is exactly at the 52-minute mark. Its angle from the 12 is:

Angle of minute hand = Number of minutes $\times$ Angle per minute

Angle of minute hand = $52 \times 6^\circ = 312^\circ$

Hour Hand Position:

At 4:52 p.m., the hour hand has moved past the 4 o'clock mark. Its position depends on both the hour (4) and the additional minutes (52).

• The hour hand starts at the 4 mark at 4:00 p.m. The angle of the 4 mark from the 12 is $4 \times 30^\circ = 120^\circ$.
• In addition to the 4 full hours, the hour hand also moves for 52 minutes within the current hour (between 4 and 5).
• The additional angle covered by the hour hand in 52 minutes is $52 \text{ minutes} \times 0.5^\circ \text{/minute} = 26^\circ$.

Total angle of the hour hand from the 12 o'clock mark:

Angle of hour hand = Angle at 4:00 + Additional angle in 52 minutes

Angle of hour hand = $120^\circ + 26^\circ = 146^\circ$

Calculating the Angle Between the Hands

The angle between the hour hand and the minute hand is the absolute difference between their positions:

Angle = $| \text{Angle of Hour Hand} - \text{Angle of Minute Hand} |$

Angle = $| 146^\circ - 312^\circ |$

Angle = $| -166^\circ |$

Angle = $166^\circ$

Finding the Smaller Angle

When two hands are on a clock face, they divide the 360 degrees of the circle into two angles. The angle we calculated ($166^\circ$) is one of these angles. The other angle is $360^\circ$ minus the calculated angle.

Other angle = $360^\circ - 166^\circ = 194^\circ$

The question asks for the smaller of the two angles. Comparing $166^\circ$ and $194^\circ$, the smaller angle is $166^\circ$.

Summary of Calculation

The smaller angle formed by the hour hand and the minute hand at 4:52 p.m. is $166^\circ$. This matches option 4.

Revision Table: Clock Angle Concepts

Additional Information: Types of Clock Problems

Clock angle problems are a common type of quantitative aptitude question. They can involve various scenarios:

• Finding the angle between hands at a given time.
• Finding the time when the hands are at a specific angle (e.g., 0 degrees for overlapping, 180 degrees for opposite).
• Calculating how many times the hands meet or are at a certain angle in a given period.
• Problems involving faulty clocks or clocks running fast or slow.

Understanding the relative speeds of the hour and minute hands is crucial for solving these problems efficiently. The minute hand gains $6^\circ - 0.5^\circ = 5.5^\circ$ on the hour hand every minute.