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.

To solve this problem, we can break it down into two right-angled triangles. The first triangle is formed by the ground station A, the balloon, and the point directly below the balloon C. The second triangle is formed by the balloon, the target B, and the point directly below the target D.

Let's start by finding the distance from the target B to the point directly below the balloon C.

In the first triangle, we have the following information:
- The length of the cable connecting the balloon to the ground station A is 842 ft.
- The angle of inclination between the cable and the horizontal is 65 degrees.
- The angle of depression of the target from the balloon is 4 degrees.

We can use trigonometry to find the height of the balloon above the point C. The trigonometric relationship we will use is tangent.

Tangent(theta) = opposite/adjacent

In triangle ABC, the opposite side is the height of the balloon (BC) and the adjacent side is the horizontal distance (AC).

Tangent(65 degrees) = BC/842 ft

Rearranging the equation, we can solve for BC:

BC = tangent(65 degrees) * 842 ft

Now, let's find the distance from the target B to the point directly below the balloon C. In the second triangle BCD, we have the following information:
- The angle of depression from the balloon to the target is 4 degrees.
- We just found the height of the balloon above the point C, which is BC.

Again, we can use trigonometry to find the distance CD. The trigonometric relationship we will use is tangent.

Tangent(theta) = opposite/adjacent

In triangle BCD, the opposite side is CD and the adjacent side is BC.

Tangent(4 degrees) = CD / BC

Rearranging the equation, we can solve for CD:

CD = tangent(4 degrees) * BC

Now, substitute the value of BC that we previously found into the equation to get CD:

CD = tangent(4 degrees) * (tangent(65 degrees) * 842 ft)

Calculating the values of tangent(4 degrees) and tangent(65 degrees), and then substituting them into the equation will give you the distance CD.

I hope this explanation helps you understand how to approach and solve the problem.

To solve this problem, we can break it down into two triangles: triangle ABC and triangle ADB.

First, let's find the length of side AB:
AB = 842 ft (the length of the cable)

Next, let's find the length of side AD:
AD = AB * sin(65 degrees)
AD = 842 ft * sin(65 degrees)
AD ≈ 753.37 ft

Now, let's find the length of side DB:
Since we have the angle of depression, we can use tan(θ) = opposite/adjacent to find DB.
tan(4 degrees) = BD/AD
BD = AD * tan(4 degrees)
BD ≈ 52.92 ft

Finally, let's find the length of side CB:
CB = AB - BD
CB = 842 ft - 52.92 ft
CB ≈ 789.08 ft

Therefore, the distance from the target B to point C directly under the balloon is approximately 789.08 ft.