Asronomy/Trig struggles..

Question

How do I solve? Step by step

When Jupiter is at a distance of 630 million kilometers from Earth (4.2 astronomical units), it has an apparent angular diameter of 46 arcseconds (46/3600 of a degree). Calculate Jupiter’s diameter in kilometers, and state how many times larger this is than the diameter of Earth

Answer 1

I usually recommend Step 1: draw a picture. Unfortunately, I can't in this program. So use the following definitions:

Let A be the angular diameter, in degrees, D the distance to the object, and d diameter of the object; D and d are in the same units.

Step 2: set up the trig formula: The ray D bisects the angle A and with d/2 are the two legs of a right triangle.

Step 3: the tangent of the angle [A / 2] = (d/2) / D ==> tan [A/2] = d / 2D

using the small angle approximation - sin A ~= A and cos A ~= 1 for angles this small, so tan A ~= A also.

Step 4: solve for d:

d = 2D * [A/2] = D*A

Step 5: Substitute in known values

d ~= (6.30 x 10⁸ km)*(46 / 3600)

Step 6: solve (which I will leave for you to do since my calculator has apparently taken a hike).

Hope this helps