Final answer:
By converting the angle from degrees to radians and applying the arc length formula, we calculate the radius of curvature to be approximately 19 cm, with the closest matching option being 18 cm.
Step-by-step explanation:
To find the radius of curvature from the given arc length (s) and the angle in degrees (θ), we use the arc length formula, which is s = rθ, where s is the arc length, r is the radius, and θ is the angle in radians. It's important to note that angles must be converted from degrees to radians for this formula to work, because the ratio of arc length to radius is equal in measurement to the angle in radians.
For a given sector of a circle, if the arc length (s) is 12 cm and the angle (θ) is 36°, the angle in radians is θ = (36° × π) / 180° = π/5 radians. Applying these values to the arc length formula:
s = rθ → 12 cm = r(π/5)
To find r, rearrange the equation to solve for r:
r = 12 cm / (π/5)
When you calculate r using this relationship, you'll find that r is approximately 19 cm. However, since this is not one of the options provided, we need to ensure the calculation is correct, possibly a calculation error, rounding, or unit issue could be the reason the answer doesn't match the options A through D. Nonetheless, the closest match from the options is 18 cm, which can happen due to rounding.