I don't know how to solve this at all. You work every Sunday in the yard from 8:00 A.M. To 11:30 A.M. Draw a diagram that shows the rotation completed by the hour hand of a clock during this time. Find the measure of the angle generated by the hour hand in both degrees and radians. Compare this angle with the angle generated by the minute hand from 8:00 A.M. To 11:30 A.M.
To draw a diagram that shows the rotation completed by the hour hand of a clock during the given time, you can follow these steps:
1. Start by drawing a circle to represent the clock face.
2. Divide the circle into 12 equal parts, representing the hours on the clock.
3. Draw a straight line from the center of the circle (representing the pivot point of the hour hand) to the 12 o'clock position.
4. Identify the starting position of the hour hand at 8:00 A.M. and mark it on the circle.
5. Draw a line connecting the center of the circle to the marked position of the hour hand.
6. Repeat steps 4 and 5 for 9:00 A.M., 10:00 A.M., and 11:30 A.M., marking the positions of the hour hand on the circle and connecting them with lines to the center.
To find the measure of the angle generated by the hour hand in both degrees and radians, use the fact that there are 12 hours on a clock, which represents a total angle of 360 degrees or 2π radians.
At 8:00 A.M., the hour hand points directly at the 8, which is 2/12 of the way around the clock. Therefore, the angle generated by the hour hand at this time is:
- In degrees: (2/12) * 360 = 60 degrees.
- In radians: (2/12) * 2π ≈ 1.047 radians.
At 11:30 A.M., the hour hand points between the 11 and the 12, which is 11.5/12 of the way around the clock. Therefore, the angle generated by the hour hand at this time is:
- In degrees: (11.5/12) * 360 ≈ 345 degrees.
- In radians: (11.5/12) * 2π ≈ 6.021 radians.
To compare this angle with the angle generated by the minute hand from 8:00 A.M. to 11:30 A.M., you need to calculate the angle generated by the minute hand during this time. Since the minute hand completes a full revolution in 60 minutes, you can calculate the fraction of the full revolution that corresponds to the time interval.
From 8:00 A.M. to 11:30 A.M., there are 3 hours and 30 minutes, or 210 minutes. Therefore, the angle generated by the minute hand during this time is:
- In degrees: (210/60) * 360 = 1260 degrees.
- In radians: (210/60) * 2π ≈ 21.991 radians.
Comparing these two angles, you can see that the angle generated by the minute hand is much larger than the angle generated by the hour hand during the same time interval.
To solve this problem, we need to understand how the hour hand of a clock moves and calculate the angle it covers during the given time period.
The hour hand of a clock completes one full rotation (360 degrees) in 12 hours, which means it moves by 360/12 = 30 degrees per hour. To find the angle covered in a specific time period, we need to determine the number of hours and minutes elapsed.
In this case, the time period is from 8:00 A.M. to 11:30 A.M., which is a total of 3 hours and 30 minutes. We need to convert the 30 minutes into hours by dividing it by 60. So, 30 minutes is equal to 30/60 = 0.5 hours.
Now, we can calculate the angle covered by the hour hand:
Angle covered by the hour hand = (Number of hours + (Number of minutes / 60)) * 30 degrees.
Therefore, the angle covered by the hour hand from 8:00 A.M. to 11:30 A.M is:
(3 + 0.5) * 30 = 3.5 * 30 = 105 degrees.
To convert this angle to radians, we need to use the conversion factor: 180 degrees = π radians.
Angle covered by the hour hand in radians = (105 degrees * π) / 180.
Now, let's calculate the angle generated by the minute hand from 8:00 A.M. to 11:30 A.M. The minute hand completes one full rotation (360 degrees) in 60 minutes, which means it moves by 360/60 = 6 degrees per minute.
The time period is 3 hours and 30 minutes, so:
Angle covered by the minute hand = (Number of hours * 60 + Number of minutes) * 6 degrees.
Therefore, the angle covered by the minute hand from 8:00 A.M. to 11:30 A.M is:
(3 * 60 + 30) * 6 = 210 * 6 = 1260 degrees.
To convert this angle to radians:
Angle covered by the minute hand in radians = (1260 degrees * π) / 180.
Now, you can compare the angles covered by the hour hand and the minute hand during the given time period.