Final answer:
To find the central angle in degrees for an arc length of 27.0 ft on a circle with a radius of 15.0 ft, we calculate it in radians first (27.0 ft / 15.0 ft = 1.8 radians) and then convert to degrees, which approximates to 103.1314 degrees. Since this value is not one of the options provided, we consider the closest option b) 108 degrees.
Step-by-step explanation:
To determine the angle at the center of a circle with radius 15.0 ft for an arc length of 27.0 ft, we use the formula which relates the arc length (Δs), the radius (r), and the central angle in radians (Δθ): Δs = rΔθ. To find the central angle in degrees, we first solve for Δθ in radians by rearranging the equation to Δθ = Δs / r. Substituting the values, we get Δθ = 27.0 ft / 15.0 ft = 1.8 radians.
Since 1 radian is approximately 57.2958 degrees, we convert the angle from radians to degrees by multiplying by this conversion factor: 1.8 radians * 57.2958 degrees/radian ≈ 103.1314 degrees. However, this is not one of the provided options. Typically, we would look for the number closest to this calculation, in this case, 108 degrees. However, the question wording suggests that we should instead provide the closest option, which would be option b) 108 degrees as the answer.