Question

Two planets are orbiting a star. Planet B can be seen from Planet A with the

eye, but as the figure shows, Planet could be located at either of two possible positions. Planet A is 65 million miles from the star and Planet B is 45 million miles from the star, as shown in the figure below. If the viewing angle between the star and Planet B is 19 , find the possible distances from Planet A to Planet B . Round your answers to the nearest tenth.

Answer 1

Explanation:

Since the viewing angle between the star and Planet B is 19 degrees, we can use trigonometry to find the distances from Planet A to Planet B. Let's call these distances x and y, where x is the distance to the closer position of Planet B and y is the distance to the farther position of Planet B, as shown in the figure.

Using the Law of Cosines for the triangles with sides 65, x, and 45 (for the closer position) and 65, y, and 45 (for the farther position), we can write:

x^2 = 65^2 + 45^2 - 26545*cos(19) ≈ 2374.6

y^2 = 65^2 + 45^2 - 26545*cos(161) ≈ 10864.6

Taking the square roots of both sides, we get:

x ≈ 48.7 and y ≈ 104.2

Therefore, the possible distances from Planet A to Planet B are approximately 48.7 million miles (for the closer position of Planet B) and 104.2 million miles (for the farther position of Planet B), rounded to the nearest tenth.