Final answer:
The pie plate moves through 183 / (10π) revolutions, which is 183/5 radians or (183 / (10π)) * 360 degrees. Each of the 10 equal slices of the pie has an angular size of π/5 radians.
Step-by-step explanation:
To calculate the angular distance the pie plate has moved through in revolutions, radians, and degrees, we first need to use the relationship between arc length, radius, and angle of rotation. The formula for the circumference of a circle, which is the arc length for one revolution, is C = 2πr, where r is the radius, and π (pi) is approximately 3.14159.
The radius of the 10-inch diameter plate is 5 inches (r = 10/2). The rim of the pie plate moving through a distance of 183 inches corresponds to Δs, the arc length in the formula Δθ = Δs/r, where Δθ is the angle of rotation in radians.
First, calculate the number of revolutions (Δθ in revolutions): 183 inches arc length divided by the circumference of the pie plate (2π * 5). Then, convert the answer to radians by multiplying by 2π radians per revolution and to degrees by multiplying by 360 degrees per revolution.
183 inches / (2π * 5 inches) = 183 / (10π) revolutions. In radians, this gives (183 / (10π)) * 2π = 183/5 radians, and in degrees, (183 / (10π)) * 360 degrees.
For the angle of one slice of the pie, since there are 10 equal slices and a full pie is 2π radians, one slice is 2π/10 = π/5 radians.