a captive balloon is connected to a ground station A by a cable of length 842 ft inclined 65 degrees to the horizontal. In a vertical plane with the balloon and its station and on the opposite side of the balloon from A, a target B was sighted from the balloon on a level with A. if the angle of depression ofthe target from the balloon is 4 degrees, find the distance from the target to a point C directly under the balloon.
The cable is the hyp. of a rt. triangle,
and the ht. of the balloon is the ver.
side.
h = 842*sin65 = 763.1 Ft. = Ht. of the balloon.
Tan4 = h/BC
BC = h/Tan4 = 763.1/Tan4 = 10,913 Ft =
Distance from the target to point C.
To solve this problem, we need to break it down into smaller steps. Let's start by visualizing the situation described.
1. Draw a diagram:
- Place a point A on the ground, representing the ground station.
- Draw a vertical line from point A, and label it as the height of the captive balloon.
- From the top of the vertical line (balloon), draw a slanting line at an angle of 65 degrees to the horizontal. Label it as the cable.
- On the opposite side of the balloon, place a point B to represent the target.
- From the balloon, draw a line downward at an angle of depression of 4 degrees. Label it as the line of sight.
B
/
/|
/ |
/ |
A---C
2. Identify what we know:
- The length of the cable (AC) is given as 842 ft.
- The angle of depression (angle BAC) is given as 4 degrees.
3. Determine what we need to find:
- We need to find the distance from the target (B) to a point directly under the balloon (C).
Now let's calculate the distance from the target to point C using trigonometry.
4. Use Trigonometry:
- In the right triangle ABC, we have the angle BAC (angle of depression) as 4 degrees, the opposite side is BC (the vertical distance we want to find), and the adjacent side is AC (given as 842 ft).
tan(angle) = opposite / adjacent
tan(4 degrees) = BC / 842 ft
- Rearranging the equation to solve for BC gives:
BC = tan(4 degrees) * 842 ft
5. Calculate BC:
- Plug the values into a calculator or use a trigonometric table to find the tangent of 4 degrees:
BC = tan(4 degrees) * 842 ft
BC ≈ 0.0698 * 842 ft
BC ≈ 58.246 ft
Therefore, the distance from the target (B) to the point directly under the balloon (C) is approximately 58.246 ft.