Calculating Clock Angles at 6:43 a.m.

The question asks for the measure of the acute angle formed between the hour hand and the minute hand at 6:43 a.m. To solve this, we need to calculate the position of both the hour hand and the minute hand relative to the 12 o'clock mark, which we consider 0 degrees.

Understanding Clock Hand Movement

A standard analog clock face is a circle of 360 degrees. There are 12 hours marked on the face.

• The minute hand completes a full circle (360 degrees) in 60 minutes.
• The hour hand completes a full circle (360 degrees) in 12 hours.

Calculating Hand Speeds

Let's determine the speed of each hand in degrees per minute:

• Minute Hand Speed: The minute hand moves \(360^\circ\) in 60 minutes. So, its speed is \(\frac{360^\circ}{60 \text{ minutes}} = 6^\circ\) per minute.
• Hour Hand Speed: The hour hand moves \(360^\circ\) in 12 hours, which is \(12 \times 60 = 720\) minutes. So, its speed is \(\frac{360^\circ}{720 \text{ minutes}} = 0.5^\circ\) per minute. Alternatively, it moves \(30^\circ\) per hour (\(360^\circ / 12\)), and since there are 60 minutes in an hour, it moves \(30^\circ / 60 = 0.5^\circ\) per minute.

Calculating Hand Positions at 6:43 a.m.

The time is 6 hours and 43 minutes past midnight (or 12 o'clock). We calculate the angle of each hand from the 12 o'clock position (0 degrees), moving clockwise.

Minute Hand Angle at 6:43 a.m.

The minute hand is at the 43-minute mark. Its angle from 12 is:

Minute hand angle = \(43 \text{ minutes} \times 6^\circ/\text{minute}\)

Minute hand angle = \(43 \times 6^\circ = 258^\circ\)

Hour Hand Angle at 6:43 a.m.

The hour hand's position depends on both the hour and the minutes past the hour. At 6:43 a.m., the hour is 6, and there are 43 minutes past 6.

The hour hand moves 30 degrees for each hour mark. At exactly 6:00, the hour hand is at \(6 \times 30^\circ = 180^\circ\) from 12.

In addition to the 6 full hours, the hour hand also moves for the 43 minutes. It moves 0.5 degrees per minute.

Movement due to minutes = \(43 \text{ minutes} \times 0.5^\circ/\text{minute}\)

Movement due to minutes = \(43 \times 0.5^\circ = 21.5^\circ\)

Total hour hand angle from 12 = Angle at 6:00 + Movement in 43 minutes

Total hour hand angle = \(180^\circ + 21.5^\circ = 201.5^\circ\)

Calculating the Angle Between the Hands

The angle between the hands is the absolute difference between their angles from the 12 o'clock position.

Angle difference = |Hour Hand Angle - Minute Hand Angle|

Angle difference = \(|201.5^\circ - 258^\circ|\)

Angle difference = \(|-56.5^\circ|\)

Angle difference = \(56.5^\circ\)

This calculation gives the direct angle between the hands. Since the result \(56.5^\circ\) is less than 180 degrees, it is the acute angle. If the calculation had resulted in an angle greater than 180 degrees, the acute angle would be \(360^\circ\) minus the calculated angle.

At 6:43 a.m., the angle between the hour hand and the minute hand is 56.5 degrees.

Revision Table: Clock Angle Calculation at 6:43 a.m.

Additional Information: Clock Angle Concepts

Calculating angles between clock hands is a common type of problem. The general formulas used are:

• Angle of hour hand from 12 = \((H \times 30) + (M \times 0.5)\), where H is the hour (0-11) and M is the minutes.
• Angle of minute hand from 12 = \(M \times 6\), where M is the minutes.
• The angle between hands = \(| \text{Hour Hand Angle} - \text{Minute Hand Angle} |\).
• If the result is greater than 180 degrees, the acute angle is \(360^\circ - \text{result}\).

Remember that the hour hand's position is not fixed at the hour mark but constantly moves as the minutes pass.