At which one of the following times, do the hour hand and the minute hand of the clock make an angle 180° with each other?

Correct Answer: Between 7:05 hours and 7:10 hours

Step-by-step Explanation:

1. Understanding the Clock Geometry:

A clock face is a circle consisting of 360°, divided into 12 equal hour spaces. Each hour space represents:

$\frac{360°}{12}=30°$

2. Relative Speed of the Hands:

• The minute hand completes a full circle (360°) in 60 minutes. Therefore, its speed is:

$\frac{360°}{60}=6°\text{per minute}$

• The hour hand moves 30° in 60 minutes. Therefore, its speed is:

$\frac{30°}{60}=0.5°\text{per minute}$

• The relative speed of the minute hand with respect to the hour hand is:

$6°-0.5°=5.5°\text{per minute}$

3. Initial Position at 7:00:

At exactly 7:00, the minute hand points at 12 and the hour hand points at 7.

The angle measuring clockwise from the minute hand (at 12) to the hour hand (at 7) is:

$7\times 30°=210°$

4. Finding the Time when the Angle is 180°:

For the hands to make a 180° angle (be in a straight line pointing in opposite directions) after 7:00, the minute hand must reduce the initial separation of 210° down to 180°.

The angular distance the minute hand needs to gain relative to the hour hand is:

$210°-180°=30°$

Let t be the time in minutes after 7:00. Using the relative speed:

$5.5\times t=30$

$t=\frac{30}{5.5}=\frac{60}{11}=5\frac{5}{11}\text{minutes}$

5. Conclusion:

The hands will be opposite to each other at 5⁵/₁₁ minutes past 7, which lies strictly Between 7:05 hours and 7:10 hours.