Final answer:
The pie plate with a 9.5-inch diameter on a rotating tray moves through an angular distance of approximately 10.325 revolutions, 64.8421 radians, and 3714.92 degrees after the rim travels 308 inches.
Step-by-step explanation:
The question asks us to express the angular distance that a pie plate with a 9.5-inch diameter has moved through in revolutions, radians, and degrees, after being on a rotating tray that has moved the rim through a distance of 308 inches. To find the angular distance, we need to use the relationship between the arc length, radius, and rotation angle.
Firstly, we calculate the radius (r) of the plate by dividing the diameter by two: r = 9.5 in / 2 = 4.75 in. The rotation angle in radians (Δθ) is given by the arc length (Δs) divided by the radius: Δθ = Δs / r = 308 in / 4.75 in = 64.8421 radians, approximately.
Since one revolution is 2π radians, we can determine the number of revolutions by dividing the angle in radians by 2π: Number of revolutions = 64.8421 radians / (2π) ≈ 10.325 revolutions.
To convert radians to degrees, we use the fact that 1 radian = 57.2958 degrees. Thus, the angle in degrees is Δθ in radians × 57.2958 degrees/radian = 64.8421 × 57.2958 ≈ 3714.92 degrees.