Final answer:
The arc length of a sector in a circle with a diameter of 12.0 inches subtended by an angle of 17°30' is calculated by first determining the radius and the circumference of the circle, converting the angle to radians, and then using the formula S = rθ to find the arc length, which is approximately 1.83 inches rounded to the nearest hundredth.
Step-by-step explanation:
To calculate the arc length of a sector in a circle with a diameter of 12.0 inches, subtended by an angle of 17°30', we use the relationship between the circumference of the circle, the fraction of the circle's rotation, and the given angle. First, we convert the diameter to the radius (r = 6.0 inches) since the radius is needed to calculate the circumference. Then, we determine the circumference (C) of the circle, which is given by the formula C = 2πr.
Next, we convert the angle of 17°30' (17.5 degrees as there are 60 minutes in a degree) to radians, as this is the standard unit in arc length calculations. One degree is approximately 0.0174533 radians, so 17.5 degrees is about 0.305432 radians. The arc length (S) will be the fraction of the circumference that corresponds to this angle, which can be found using the formula S = rθ, where θ is the angle in radians.
Plugging in our values, we get S = 6.0 inches × 0.305432 radians, which results in an arc length S = 1.83 inches (rounded to the nearest hundredth).