Answer :
Absolutely! Let’s find the smaller angles between the hour hand and the minute hand of a clock at the specified times. Here's the question:
### Problem:
Find the smaller angles between the hour hand and the minute hand of a clock at the following times. Give your answers in degrees:
a) [tex]\(\frac{1}{2}\)[/tex] past 3 (i.e., 3:30)
b) [tex]\(\frac{1}{2}\)[/tex] past 10 (i.e., 10:30)
### Solution:
#### Part (a) [tex]\(\frac{1}{2}\)[/tex] past 3 (3:30)
1. Hour Hand Position:
- The hour hand is at 3:30.
- An hour hand moves [tex]\(30\)[/tex] degrees per hour (since [tex]\(360^\circ\)[/tex] for a full clock divided by [tex]\(12\)[/tex] hours is [tex]\(30^\circ\)[/tex] per hour).
- At 3:00, the hour hand is at [tex]\(3 \times 30 = 90\)[/tex] degrees.
- At 3:30, the hour hand has moved halfway to 4:00. In [tex]\(\frac{1}{2}\)[/tex] hour, it moves an extra [tex]\(15\)[/tex] degrees (since [tex]\(30^\circ \times 0.5 = 15^\circ\)[/tex]).
- Therefore, at 3:30, the hour hand is at [tex]\(90 + 15 = 105\)[/tex] degrees from the 12 o’clock position.
2. Minute Hand Position:
- The minute hand is at 30 minutes.
- A minute hand moves [tex]\(6\)[/tex] degrees per minute (since [tex]\(360^\circ\)[/tex] for a full clock divided by [tex]\(60\)[/tex] minutes is [tex]\(6^\circ\)[/tex] per minute).
- Therefore, at 30 minutes, the minute hand is at [tex]\(30 \times 6 = 180\)[/tex] degrees from the 12 o’clock position.
3. Angle Between Hour and Minute Hands:
- The angle between the hour hand and minute hand is the absolute difference between their positions.
- Angle [tex]\(= |105 - 180| = 75\)[/tex] degrees.
- Since this angle is less than [tex]\(180\)[/tex] degrees, it is the smaller angle.
So, the smaller angle at [tex]\(\frac{1}{2}\)[/tex] past 3 is [tex]\(75\)[/tex] degrees.
#### Part (b) [tex]\(\frac{1}{2}\)[/tex] past 10 (10:30)
1. Hour Hand Position:
- The hour hand is at 10:30.
- At 10:00, the hour hand is at [tex]\(10 \times 30 = 300\)[/tex] degrees from the 12 o’clock position.
- At 10:30, the hour hand has moved halfway to 11:00. In [tex]\(\frac{1}{2}\)[/tex] hour, it moves an extra [tex]\(15\)[/tex] degrees.
- Therefore, at 10:30, the hour hand is at [tex]\(300 + 15 = 315\)[/tex] degrees from the 12 o’clock position.
2. Minute Hand Position:
- The minute hand is at 30 minutes.
- Therefore, at 30 minutes, the minute hand is at [tex]\(180\)[/tex] degrees from the 12 o’clock position.
3. Angle Between Hour and Minute Hands:
- The angle between the hour hand and minute hand is the absolute difference between their positions.
- Angle [tex]\(= |315 - 180| = 135\)[/tex] degrees.
- Since this angle is less than [tex]\(180\)[/tex] degrees, it is the smaller angle.
So, the smaller angle at [tex]\(\frac{1}{2}\)[/tex] past 10 is [tex]\(135\)[/tex] degrees.
### Summary:
a) The smaller angle at [tex]\(\frac{1}{2}\)[/tex] past 3 (3:30) is [tex]\(75\)[/tex] degrees.
b) The smaller angle at [tex]\(\frac{1}{2}\)[/tex] past 10 (10:30) is [tex]\(135\)[/tex] degrees.
### Problem:
Find the smaller angles between the hour hand and the minute hand of a clock at the following times. Give your answers in degrees:
a) [tex]\(\frac{1}{2}\)[/tex] past 3 (i.e., 3:30)
b) [tex]\(\frac{1}{2}\)[/tex] past 10 (i.e., 10:30)
### Solution:
#### Part (a) [tex]\(\frac{1}{2}\)[/tex] past 3 (3:30)
1. Hour Hand Position:
- The hour hand is at 3:30.
- An hour hand moves [tex]\(30\)[/tex] degrees per hour (since [tex]\(360^\circ\)[/tex] for a full clock divided by [tex]\(12\)[/tex] hours is [tex]\(30^\circ\)[/tex] per hour).
- At 3:00, the hour hand is at [tex]\(3 \times 30 = 90\)[/tex] degrees.
- At 3:30, the hour hand has moved halfway to 4:00. In [tex]\(\frac{1}{2}\)[/tex] hour, it moves an extra [tex]\(15\)[/tex] degrees (since [tex]\(30^\circ \times 0.5 = 15^\circ\)[/tex]).
- Therefore, at 3:30, the hour hand is at [tex]\(90 + 15 = 105\)[/tex] degrees from the 12 o’clock position.
2. Minute Hand Position:
- The minute hand is at 30 minutes.
- A minute hand moves [tex]\(6\)[/tex] degrees per minute (since [tex]\(360^\circ\)[/tex] for a full clock divided by [tex]\(60\)[/tex] minutes is [tex]\(6^\circ\)[/tex] per minute).
- Therefore, at 30 minutes, the minute hand is at [tex]\(30 \times 6 = 180\)[/tex] degrees from the 12 o’clock position.
3. Angle Between Hour and Minute Hands:
- The angle between the hour hand and minute hand is the absolute difference between their positions.
- Angle [tex]\(= |105 - 180| = 75\)[/tex] degrees.
- Since this angle is less than [tex]\(180\)[/tex] degrees, it is the smaller angle.
So, the smaller angle at [tex]\(\frac{1}{2}\)[/tex] past 3 is [tex]\(75\)[/tex] degrees.
#### Part (b) [tex]\(\frac{1}{2}\)[/tex] past 10 (10:30)
1. Hour Hand Position:
- The hour hand is at 10:30.
- At 10:00, the hour hand is at [tex]\(10 \times 30 = 300\)[/tex] degrees from the 12 o’clock position.
- At 10:30, the hour hand has moved halfway to 11:00. In [tex]\(\frac{1}{2}\)[/tex] hour, it moves an extra [tex]\(15\)[/tex] degrees.
- Therefore, at 10:30, the hour hand is at [tex]\(300 + 15 = 315\)[/tex] degrees from the 12 o’clock position.
2. Minute Hand Position:
- The minute hand is at 30 minutes.
- Therefore, at 30 minutes, the minute hand is at [tex]\(180\)[/tex] degrees from the 12 o’clock position.
3. Angle Between Hour and Minute Hands:
- The angle between the hour hand and minute hand is the absolute difference between their positions.
- Angle [tex]\(= |315 - 180| = 135\)[/tex] degrees.
- Since this angle is less than [tex]\(180\)[/tex] degrees, it is the smaller angle.
So, the smaller angle at [tex]\(\frac{1}{2}\)[/tex] past 10 is [tex]\(135\)[/tex] degrees.
### Summary:
a) The smaller angle at [tex]\(\frac{1}{2}\)[/tex] past 3 (3:30) is [tex]\(75\)[/tex] degrees.
b) The smaller angle at [tex]\(\frac{1}{2}\)[/tex] past 10 (10:30) is [tex]\(135\)[/tex] degrees.
