The restaurant in the Space Needle in Seattle rotates at the rate of one revolution per hour.
(a) Through how many radians does it turn in 100 minutes?
(b) How long does it take the restaurant to rotate through 4 radians?
(c) How far does a person sitting by the window move in 100 minutes if the radius of the restaurant is 21 meters?
actually, i managed to figure it out! i no longer need help :)
(a) 2π * 100/60 = ____
(b) 4rad * 1hr/2πrad = 2/π hr
(c) s = rθ = 21 * (a)
(a) To find the number of radians the restaurant turns in 100 minutes, we can use the formula:
Number of radians = (rate of rotation in rev/hr) * (time in minutes) * (2π radians/revolution)
The rate of rotation is 1 revolution per hour, which is equal to 2π radians.
Plugging in the values:
Number of radians = (2π radians/1 hour) * (100 minutes/60 minutes) = (2π radians) * (5/3) = 10π/3 radians.
Therefore, the restaurant turns through 10π/3 radians in 100 minutes.
(b) To find the time it takes for the restaurant to rotate through 4 radians, we can rearrange the formula:
Time = (Number of radians) / ((rate of rotation in rev/hr) * (2π radians/revolution))
Plugging in the values:
Time = 4 radians / ((1 revolution/1 hour) * (2π radians/revolution)) = 4 / (2π) hours = 2 / π hours.
Therefore, the restaurant takes 2 / π hours to rotate through 4 radians.
(c) To find how far a person sitting by the window moves in 100 minutes, we can use the formula:
Distance = (rate of rotation in rev/hr) * (time in minutes) * (2π radius)
The rate of rotation is 1 revolution per hour, which is equal to 2π radians.
Plugging in the values:
Distance = (2π radians/1 hour) * (100 minutes/60 minutes) * (21 meters) = (2π radians) * (5/3) * (21 meters) = 35π meters.
Therefore, a person sitting by the window moves 35π meters in 100 minutes.
To answer these questions, we need to understand the relationship between radians, time, and angular velocity.
(a) To find out how many radians the restaurant turns in 100 minutes, we first convert the given time to hours. Since there are 60 minutes in an hour, 100 minutes is equal to 100/60 = 5/3 hours.
Next, we know that the restaurant completes one revolution per hour. Since a full revolution is equal to 2π radians, the restaurant rotates at an angular velocity of 2π radians per hour.
Given that the restaurant rotates for 5/3 hours, we can calculate the number of radians using the formula:
Number of radians = (angular velocity) x (time in hours)
Number of radians = 2π radians/hour x (5/3) hours
Number of radians = (10π/3) radians
Therefore, the restaurant turns through (10π/3) radians in 100 minutes.
(b) To determine the time it takes for the restaurant to rotate through 4 radians, we use the same formula:
Time in hours = (Number of radians) / (angular velocity)
Given that the number of radians is 4 and the angular velocity is 2π radians/hour, we can substitute these values into the formula:
Time in hours = 4 radians / (2π radians/hour)
Time in hours = 2 hours
Thus, the restaurant takes 2 hours to rotate through 4 radians.
(c) To find out how far a person sitting by the window moves in 100 minutes, we need to calculate the arc length. The arc length is the distance traveled along the circular path.
The formula for arc length is given by:
Arc length = (angle in radians) x (radius)
Substituting the given values, we have:
Arc length = (10π/3) radians x 21 meters
Arc length ≈ 70π meters
Therefore, a person sitting by the window moves approximately 70π meters in 100 minutes.