the minute hand of a clock is 9 cm and that of hour hand is 7 cm.If they ran for 3 days,how much distance is covered by both the hands?
how many times around does each hand go?
Each time around has circumference 2πr for the hand's radius.
Well, if the minute hand and the hour hand went for a run, they might end up in a clock-a-thon! But I'm afraid I can't calculate the distance covered by the hands, because it's not like they're running track and field. They're just pointing at numbers on the clock. So, instead of distance, let's just say they covered the full 12-hour rotation multiple times over the course of three days. That's a lot of rounds!
To find the distance covered by both the minute and hour hands of a clock over a period of time, we need to consider their respective speeds and the duration of time.
1. Calculate the circumference of the clock face:
The circumference of a circle is given by the formula: C = 2πr, where r is the radius.
Given that the minute hand has a length of 9 cm, the radius (r) would be 9 cm.
Therefore, the circumference of the clock face is:
C = 2 * π * 9 cm = 18π cm.
2. Calculate the speed of the minute hand:
The minute hand of a clock completes a full rotation in 60 minutes.
Therefore, in 1 minute, the minute hand covers a distance equal to the circumference of the clock face.
So, the speed of the minute hand is: 18π cm/minute.
3. Calculate the speed of the hour hand:
The hour hand of a clock completes a full rotation in 12 hours.
Therefore, in 1 hour (60 minutes), the hour hand covers a distance equal to the circumference of the clock face.
So, the speed of the hour hand is: 18π cm/hour.
4. Calculate the total distance covered by both hands over 3 days:
Since there are 24 hours in a day and 60 minutes in an hour, the total time in minutes can be calculated as:
Total time = 3 days * 24 hours/day * 60 minutes/hour = 4320 minutes.
The distance covered by the minute hand over the 3-day period is:
Distance = Speed * Time = (18π cm/minute) * 4320 minutes.
The distance covered by the hour hand over the 3-day period is:
Distance = Speed * Time = (18π cm/hour) * 4320 minutes.
5. Calculate the total distance covered by both hands:
Total distance = Distance covered by minute hand + Distance covered by hour hand.
Now, let's calculate the total distance covered by both the minute and hour hands over 3 days.
To find the distance covered by both the minute and hour hands of a clock, we need to consider their respective speeds and the amount of time they run for.
First, let's determine the speed at which each hand moves. The minute hand completes one revolution in 60 minutes (1 hour), and the hour hand completes one revolution in 12 hours.
The length of the minute hand is given as 9 cm, which represents the distance covered in one full revolution. Therefore, the speed of the minute hand is 9 cm per 60 minutes (or 1 hour), which simplifies to 0.15 cm per minute.
Similarly, the length of the hour hand is given as 7 cm, representing the distance covered in one full revolution. So, the speed of the hour hand is 7 cm per 12 hours, which simplifies to 0.58 cm per hour (or 0.0097 cm per minute).
Now, let's calculate the total distance covered by each hand over the course of 3 days:
Distance covered by the minute hand:
Speed of the minute hand = 0.15 cm per minute
Time in 3 days = 3 days × 24 hours × 60 minutes = 4320 minutes
Distance covered by the minute hand = 0.15 cm per minute × 4320 minutes = 648 cm
Distance covered by the hour hand:
Speed of the hour hand = 0.0097 cm per minute
Time in 3 days = 3 days × 24 hours × 60 minutes = 4320 minutes
Distance covered by the hour hand = 0.0097 cm per minute × 4320 minutes = 41.9 cm
Therefore, the total distance covered by both the minute and hour hands in 3 days is 648 cm + 41.9 cm = 689.9 cm, or approximately 689.9 centimeters.