A marble has a diameter of 2 cm. At what distance would the marble have an angular diameter of 1 arc second
A)4.1 cm
B)4.1 miles
C)4.1 m
D)4.1 ft
E)4.1 km

Question

Quizplus · Accepted Answer

The angular diameter, $ \delta $, in radians can be calculated using the formula $ \delta = \frac{d}{D} $, where $ d $ is the actual diameter of the object and $ D $ is the distance from the observer. To convert 1 arc second to radians, we use the conversion $ 1 \, \text{arc second} = \frac{1}{3600} \times \frac{\pi}{180} $ radians. Given that the diameter of the marble is 2 cm, we want to find the distance $ D $ that results in an angular diameter of 1 arc second. $$ 1 \, \text{arc second} = \frac{1}{3600} \times \frac{\pi}{180} \, \text{radians} = \frac{\pi}{648000} \, \text{radians} $$$$ \frac{\pi}{648000} = \frac{2 \, \text{cm}}{D} $$$$ D = \frac{2 \, \text{cm} \times 648000}{\pi} $$$$ D \approx 412316.861 \, \text{cm} $$$$ D \approx 4.123 \, \text{km} $$Therefore, the marble would have an angular diameter of 1 arc second at a distance of approximately 4.1 kilometers.

A Marble Has a Diameter of 2 Cm