Final answer:
The width of the road sign at 1 km away and with an angular size of 120 arc seconds is approximately 0.589 meters after converting the angular size to radians and applying the small angle approximation formula. The given options do not match this result, suggesting there may be an issue with the initial question or options provided. thus, the correct option is D).
Step-by-step explanation:
To find out the width of a road sign that has an angular size of 120 arc seconds from 1 km away, we need to use a small angle approximation formula where the angle is given in radians and the distance in the same unit of length for the size we want to obtain. First, we have to convert 120 arc seconds to radians.
An arc second is 1/3600 of a degree, and there are π radians in 180 degrees, so to convert 120 arc seconds to radians:
120 arc seconds * (1 degree / 3600 arc seconds) * (π radians / 180 degrees) = (120/3600) * (π/180) radians
This equals about 5.89 × 10⁻´ radians.
Using the formula for the linear size D = θr, where θ is the angle in radians and r is the distance, and plugging in our values we have:
D = 5.89 × 10⁻´ radians * 1,000 meters
The sign is therefore approximately 0.589 meters across, which is not one of the provided options, indicating a potential error in either the given options or in the initial question.