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:

Let's call the distance from Planet A to one of the possible positions of Planet B "x". We can use trigonometry to relate the angle between the star and Planet B (19°), the distance from the star to Planet A (65 million miles), and the distances from Planet A to Planet B (x).

In a right triangle with angle 19°, the opposite side is x and the adjacent side is 65 - 45 = 20 million miles. Therefore, we can use the tangent function to relate these sides:

tan(19°) = x / 20

Solving for x, we get:

x = 20 tan(19°) ≈ 6.7 million miles

So one possible distance from Planet A to Planet B is approximately 6.7 million miles.

However, there is another possible position for Planet B, which is on the other side of Planet A as shown in the figure. In this case, the distance from Planet A to Planet B is:

y = 65 + 45 + 20 = 130 million miles

Therefore, the two possible distances from Planet A to Planet B are approximately 6.7 million miles and 130 million miles (rounded to the nearest tenth).