To determine the angle between the minute and hour hand of a clock at 5:45, we need to follow a systematic approach. Let’s break down the process step by step and use the provided formulas for accuracy.

The Angle Calculation Process

First, let’s understand the movement of the clock hands over time. The minute hand moves 360 degrees in 60 minutes, and the hour hand moves 360 degrees in 12 hours, which translates to:

The minute hand moves 6 degrees per minute. The hour hand moves 30 degrees per hour, or 0.5 degrees per minute.

Given that the current time is 5:45, we can calculate the positions of the minute and hour hands step by step.

Position of the Minute Hand

The minute hand is at 45 minutes. To find its position in degrees:

Minute hand position 45 minutes * 6 degrees/minute 270 degrees.

Position of the Hour Hand

At 5:00, the hour hand is at:

5 hours * 30 degrees/hour 150 degrees.

Between 5:00 and 5:45, the hour hand moves further. To find the additional movement in degrees:

Additional movement 45 minutes * (30 degrees/hour ÷ 60 minutes) 22.5 degrees.

Total position of the hour hand at 5:45:

150 degrees 22.5 degrees 172.5 degrees.

Calculating the Angle

The angle between the minute and hour hands is calculated as the absolute difference between their positions:

Angle |Minute hand position - Hour hand position| Angle |270 degrees - 172.5 degrees| 97.5 degrees.

Therefore, the angle between the minute and hour hand at 5:45 is 97.5 degrees.

Additional Methodology

For a more general formula, we can use the following:

Angle between hour hand and minute hand formula: |30h - 5.5m|, where h is the hour and m is the minute. At 5:45, h 5 and m 45.

Substituting the values:

Angle |30 * 5 - 5.5 * 45| |150 - 247.5| 97.5 degrees.

Alternate Calculation for 5:30

For 5:30:

Minute hand position 30 minutes * 6 degrees/minute 180 degrees. Hour hand position 5 hours * 30 degrees/hour 30 minutes * 0.5 degrees/minute 150 15 165 degrees. Angle |180 - 165| 15 degrees.

Thus, the angle between the minute and hour hand at 5:30 is 15 degrees.

Factors and Formulas

Understanding the factors and formulas helps in solving such problems efficiently:

The minute hand moves 360 degrees in 60 minutes, or 6 degrees per minute. The hour hand moves 30 degrees in 60 minutes, or 0.5 degrees per minute. The formula: |30h - 5.5m| can be used to find the angle between the two hands of a clock at any given time. Adding up the degrees for the hour hand: (30 times text{hour} 0.5 times text{minute}). Understanding the position of the hand relative to the numbers on the clock helps in calculating the angles accurately.

Conclusion

By utilizing the step-by-step process and formulas, we can accurately determine the angle between the minute and hour hand of a clock. This method ensures a precise understanding and application of the concepts involved in clock angle calculation. Whether it is 5:45 or 5:30, the techniques remain consistent, making the calculation straightforward and accurate.