A grandfather clock shows a time of 7 o’clock. What is the exact radian measure of the angle between the hour hand and the minute hand?
how is it calculated?*
1 rotation = 360° = 2π radians
at 7:00 the minute hand is on the 12 (straight up) and the hour hand has moved 7/12 of a rotation
(7/12)(2π)
= 7π/6 radians ---> or 210°
so the smaller angle between the two hands
= 2π - 7π/6
= 5π/6 radians ------> 150°
bless u
To calculate the exact radian measure of the angle between the hour hand and the minute hand on a grandfather clock, you can use the following steps:
1. Determine the time on the clock.
In this case, the time is 7 o'clock.
2. Convert the time to an angle.
The hour hand of a clock completes a full revolution (360 degrees) in 12 hours. So, for every hour, it moves 360 degrees / 12 = 30 degrees.
At 7 o'clock, the hour hand would have moved 7 hours * 30 degrees/hour = 210 degrees.
The minute hand, on the other hand, completes a full revolution in 60 minutes or 360 degrees in 60 minutes. Therefore, for every minute, it moves 360 degrees / 60 = 6 degrees.
Since there are 60 minutes in an hour, at 7 o'clock, the minute hand would have moved 6 degrees/minute * 60 minutes = 360 degrees.
3. Find the difference in angles.
Now, subtract the hour hand's angle from the minute hand's angle to find the angle between them.
Angle between the hands = Minute hand angle - Hour hand angle
Angle between the hands = 360 degrees - 210 degrees
Angle between the hands = 150 degrees
4. Convert the angle from degrees to radians.
To convert degrees to radians, you can use the formula: radian = degree * π / 180.
Angle in radians = 150 degrees * π / 180
Angle in radians ≈ 2.618 radians
Therefore, the exact radian measure of the angle between the hour hand and the minute hand at 7 o'clock is approximately 2.618 radians.
Ah, time to put on my clown nose and get silly with some math! Now, to find the exact radian measure of the angle between the hour hand and the minute hand, we need to do a little time-traveling, so hop on the laughter express!
First, let's break down the problem. At 7 o'clock, the hour hand points directly at the 7, while the minute hand is pointing at the 12. Now, the hour hand rotates 360 degrees in 12 hours.
But hold your nose, here comes the funny part! We're going to calculate the fraction of the hour that has passed since the clock struck 7 o'clock. Since 7 o'clock is exactly one-third of the way between the 6 and 8, we'll use that as our reference.
So, one-third of an hour is 20 minutes (60 minutes divided by 3), which means the minute hand has moved 20 minutes past the 12. Now, a full circle is 2π radians, and a minute hand rotates 360 degrees in 60 minutes. So, the minute hand moves 2π radians in 60 minutes, which simplifies to π/30 radians in 1 minute.
To find the angle between the hour hand and the minute hand, we need to find the difference in their positions. The hour hand moves 360 degrees (2π radians) in 12 hours, which is equivalent to 720 minutes. Therefore, the hour hand moves π/30 radians in 1 minute.
Now, let's calculate the difference. The minute hand has moved π/30 radians in 20 minutes, while the hour hand has moved π/30 radians in 720 minutes. The difference is π/30 radians * (720 - 20) minutes, which simplifies to π * 700/30 radians.
Simplifying a little more, we get 70π/3 radians as the exact radian measure of the angle between the hour hand and the minute hand at 7 o'clock. Voila! Now that's some math that's worth clowning around for!
Remember, math and laughter go hand in hand, so keep on giggling!