Question

What is the angular diameter (in arcseconds) of an object that has a linear diameter of 75 cm and a distance of 2 km?

Answer 1

Answer:

The angular diameter is 77.35 arc-seconds.

Explanation:

The angular diameter, as shown in the figure, is the angle $x$  subtended by the the diameter of the object.

Before we do the calculation, we first convert everything to meters.

The diameter of the of the object in meters is

75cm =0.75m,

and the distance to the object in meters is

2 km = 2000 m.

Now, from trigonometry we get:

$tan (\frac{x}{2} )= \dfrac{radius}{length} = \dfrac{0.75/2}{2000} \\\\\dfrac{x}{2} = tan^{-1}(\dfrac{0.75/2}{2000})\\\\\dfrac{x}{2}= 0.0107\\\\\boxed{x= 0.0215^o}$

and since 1 degree = 3600 arc-seconds, $x$ in arc-seconds is

$x= 0.0215*3600 \\\\\boxed{x= 77.35''}$