To calculate the angular distance that the sweet potato pie plate has moved through, we can use the formula:
Angular Distance = Distance Traveled / Radius
Given that the rim of the plate moves through a distance of 183 inches, and the diameter of the plate is 10.50 inches, we can calculate the radius of the plate:
Radius = Diameter / 2
Radius = 10.50 / 2
Radius = 5.25 inches
Now we can calculate the angular distance in revolutions, radians, and degrees.
1. Angular distance in revolutions:
To convert the distance traveled into revolutions, we need to find the circumference of the circular path.
Circumference = 2 * π * Radius
Circumference = 2 * 3.14 * 5.25
Circumference ≈ 32.94 inches
Angular Distance (in revolutions) = Distance Traveled / Circumference
Angular Distance (in revolutions) = 183 / 32.94
Angular Distance (in revolutions) ≈ 5.55 revolutions
2. Angular distance in radians:
To convert the angular distance from revolutions to radians, we need to multiply by 2π (since there are 2π radians in one revolution).
Angular Distance (in radians) = Angular Distance (in revolutions) * 2π
Angular Distance (in radians) ≈ 5.55 * 2 * 3.14
Angular Distance (in radians) ≈ 34.78 radians
3. Angular distance in degrees:
To convert the angular distance from radians to degrees, we need to multiply by 180/π (since there are 180 degrees in one π radians).
Angular Distance (in degrees) = Angular Distance (in radians) * (180/π)
Angular Distance (in degrees) ≈ 34.78 * (180/3.14)
Angular Distance (in degrees) ≈ 1995.54 degrees
If the sweet potato pie is cut into 10 equal slices, the angular size of one slice can be calculated by dividing the angular distance of the pie plate by the number of slices.
Angular Size (in radians) = Angular Distance (in radians) / Number of Slices
Angular Size (in radians) ≈ 34.78 / 10
Angular Size (in radians) ≈ 3.48 radians
Therefore, the angular size of one slice is approximately 3.48 radians, which can be written as 3.48/π when expressed as a fraction of π.