Question

From a balloon 915 feet high, the angle of depression to the ranger headquarters is 85°14'. How far is the headquarters from a point on the ground directly below the balloon (to the nearest foot)?

Option 1: 66 ft
Option 2: 76 ft
Option 3: 81 ft
Option 4: 87 ft

Answer 1

The distance from the point on the ground directly below the balloon to the ranger headquarters is approximately 76 feet, which is Option 2. Trigonometry is used to find this distance by using the tangent function and the angle of depression from the balloon.

Step-by-step explanation:

To find the distance from the ground point directly below the balloon to the ranger headquarters, we need to use trigonometry. Given the altitude of the balloon as 915 feet and the angle of depression as 85°14', we will use the tangent function, as this angle refers to the angle between the line of sight to the headquarters and the horizontal.The tangent of an angle in a right triangle equals the length of the opposite side divided by the length of the adjacent side. In this case, the tangent of the angle of depression will be the altitude of the balloon (opposite side) divided by the unknown distance to the headquarters (adjacent side).

First, we convert the angle to decimal degrees:
85°14' = 85 + (14/60) ≈ 85.2333 degrees.
Then, we use the identity:
tan(85.2333°) = opposite/adjacent
adjacent = opposite / tan(85.2333°).
Since the opposite side is the height of the balloon, 915 feet, we have:
adjacent = 915 / tan(85.2333°).

The calculated distance is approximately 76 feet, which makes Option 2: 76 ft the correct answer.