I'm having trouble with this trig application. The scenario goes:
A ferris wheel has a diameter of approximately 65 meters. Assume it takes 110 seconds for the ferris wheel to make one complete rotation. find the angular speed of the ferris wheel in radians per minute and linear speed of the ferris wheel in miles per hour. Show work so I can understand.
I know that
1 mile = 1609 meters
s= r θ
diameter is 65 meters
the angular speed is clearly
2πrad/110s = π/55 rad/s * 60s/min = 60π/55 = 12π/11 rad/min
the linear speed is thus
2π(65/2)m/rev * 1rev/110s = 130π/220 = 13π/22 m/s
I assume you can tackle converting that to mi/hr
To find the angular speed of the ferris wheel in radians per minute, we need to use the formula:
Angular speed = θ / t
where θ is the angle (in radians) covered by the ferris wheel in a given time, and t is the time taken to cover that angle.
In this case, we know that it takes 110 seconds for the ferris wheel to make one complete rotation, which means it covers an angle of 2π radians. Therefore, θ = 2π radians and t = 110 seconds.
Plugging these values into the formula:
Angular speed = 2π / 110
Simplifying the expression:
Angular speed = π / 55
Therefore, the angular speed of the ferris wheel is π / 55 radians per minute.
Now, let's find the linear speed of the ferris wheel in miles per hour.
The linear speed of an object can be calculated using the formula:
Linear speed = Distance / Time
In this case, the distance covered by the ferris wheel in one complete rotation is equal to its circumference. The diameter of the ferris wheel is given as 65 meters, so the radius (r) is half of that, which is 32.5 meters.
The circumference (C) of a circle can be calculated using the formula:
C = 2πr
Plugging in the values:
C = 2π(32.5) = 65π meters
We want to convert this distance to miles, so we need to divide it by 1609 (since 1 mile is equal to 1609 meters):
C in miles = (65π) / 1609
Next, we need to find the time it takes for the ferris wheel to complete one rotation in hours. We know that it takes 110 seconds, so the time in hours (t) is:
t = 110 / 3600
Now, we can plug the distance (in miles) and time (in hours) values into the formula for linear speed:
Linear speed = (C in miles) / t
Plugging in the values:
Linear speed = [(65π) / 1609] / (110 / 3600)
Simplifying the expression:
Linear speed = [(65π) / 1609] * (3600 / 110)
Linear speed ≈ 3.490 miles per hour
Therefore, the linear speed of the ferris wheel is approximately 3.490 miles per hour.