From a hot air balloon, the angle between a radio antenna straight below and the base of the library downtown is 63°, as shown below. If the distance between the radio antenna and the library is 24 miles, how many miles high is the balloon? Show your work or explain how you got your answer.

Answer 1

To find out how high the balloon is, we can use trigonometry and set up a right triangle with the radio antenna as the bottom vertex, the library as the top vertex, and the balloon as the point on the hypotenuse.

Let's denote the height of the balloon as 'h'. The angle between the library and the antenna is 63°, so the angle between the antenna and the height of the balloon is also 63° (since the angle between the hypotenuse and one of the legs of the right triangle is equal to the angle between the hypotenuse and the other leg).

Using the tangent function, we can set up the following equation:

tan(63°) = h / 24 miles

By solving for 'h', we get:

h = 24 miles * tan(63°)
h ≈ 24 miles * 2.05
h ≈ 49.2 miles

Therefore, the balloon is approximately 49.2 miles high.