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.
24/x
x = 24/tan 17°
x = 12 miles
The height of the balloon is 12 miles.
To find the height of the balloon, we can form a right triangle with the distance between the radio antenna and the library as one leg and the height of the balloon as the other leg. The angle between the radio antenna straight below and the base of the library downtown is 63 degrees. This angle is complementary to the angle formed by the height of the balloon and the distance between the radio antenna and the library. Therefore, the angle between the height of the balloon and the ground is 90 - 63 = 27 degrees.
Now, we can use the tangent function to find the height of the balloon:
tan(27) = height/24
Height = 24 * tan(27)
Height = 12 miles
Therefore, the height of the balloon is 12 miles.