A carousel has a diameter of 10 m. How far (path length) does little Joey, strapped in at the very edge of the carousel, travel if the carousel is rotated through angle of 45 degrees, 45 rad and 45 rev, respectively?
circumference = 2 pi R = pi D
= 31.4 meters
45 deg = 360/8 so 31.4/8 meters
45 rad = 45/(2 pi) circles = 7.162 circles
7.162 * 31.4 = 225 meters
45 circles
45 * 31.4 = 1413 meters
To find the path length that little Joey travels on the carousel, we need to determine the distance around the circumference of the circular path. We can then use this distance to calculate the path length for each rotation angle.
First, let's find the circumference of the carousel. We know that the diameter is 10 meters, and the formula for the circumference is C = πd, where C represents the circumference and d represents the diameter.
So, the circumference of the carousel is:
C = π * 10 = 31.4 meters (rounded to one decimal place)
Now, let's calculate the path length for each rotation angle.
1. For 45 degrees:
To calculate the path length for 45 degrees, we need to find what portion of the full circle it represents. Since there are 360 degrees in a full circle, the fraction of the circle covered by 45 degrees is 45/360 = 1/8.
Path length for 45 degrees = Fraction of circle covered * Circumference
Path length for 45 degrees = (1/8) * 31.4
Path length for 45 degrees ≈ 3.925 meters (rounded to three decimal places)
2. For 45 radians:
To calculate the path length for 45 radians, we can use the formula S = rθ, where S represents the arc length, r represents the radius, and θ represents the angle in radians.
Since the radius of the carousel is half of its diameter, r = 10/2 = 5 meters.
Path length for 45 radians = 5 * 45 = 225 meters
3. For 45 revolutions:
To calculate the path length for 45 revolutions, we need to multiply the number of revolutions by the circumference.
Path length for 45 revolutions = 45 * 31.4 = 1413 meters
Therefore, the path length that little Joey travels on the carousel for each rotation angle is approximately:
- 3.925 meters for 45 degrees
- 225 meters for 45 radians
- 1413 meters for 45 revolutions