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 find the distance from the target to point C directly under the balloon, we can use trigonometry.
Let's break down the given information:
- The cable connecting the balloon to ground station A has a length of 842 ft and is inclined at an angle of 65 degrees to the horizontal.
- The angle of depression of the target B from the balloon is 4 degrees.
To solve this problem, we need to find the horizontal distance from the target B to the point C directly under the balloon.
First, let's find the vertical distance from the balloon to the target. We can use the tangent of the angle of depression:
tan(4 degrees) = vertical distance / 842 ft
vertical distance = tan(4 degrees) * 842 ft
Next, we can find the horizontal distance from the target B to the point C. Since the cable connecting the balloon to ground station A is inclined at an angle of 65 degrees, the angle between the cable and the horizontal is 90 - 65 = 25 degrees.
Using trigonometry, we can find the horizontal distance from the target B to point C:
horizontal distance = vertical distance / tan(25 degrees)
Now, let's calculate the values:
vertical distance = tan(4 degrees) * 842 ft
horizontal distance = vertical distance / tan(25 degrees)
Calculating the values:
vertical distance = tan(4 degrees) * 842 ft ≈ 58.40 ft
horizontal distance = vertical distance / tan(25 degrees) ≈ 137.39 ft
Therefore, the distance from the target B to the point C directly under the balloon is approximately 137.39 ft.
To find the distance from the target B to point C directly under the balloon, we can begin by drawing a diagram to visualize the given information. Let's break down the problem step by step:
Step 1: Identify the given information.
- The length of the cable (AC) connecting the balloon to the ground station is 842 ft.
- The angle between the cable and the horizontal (angle DAC) is 65 degrees.
- The angle of depression from the balloon to the target B is 4 degrees.
Step 2: Analyze the diagram.
In the diagram, we have a right triangle ABC where:
- Side AC represents the length of the cable connecting the balloon to the ground station (842 ft).
- Angle DAC is given as 65 degrees.
- Angle BAC is the complement of 65 degrees, which is 90 - 65 = 25 degrees.
- Angle ACD is the complement of 25 degrees, which is 90 - 25 = 65 degrees.
- Angle BCA is the angle of depression from the balloon to the target B, given as 4 degrees.
- Side BC represents the distance from the target B to the point directly beneath the balloon, which we need to find.
Step 3: Apply trigonometry.
Since we know the opposite and adjacent sides of angle BCA, we can use the tangent function to find the distance BC.
Using the tangent function: tan(angle) = opposite/adjacent
In this case:
tan(angle BCA) = BC / AC
Plugging in the values:
tan(4 degrees) = BC / 842 ft
Now, we can solve for BC:
BC = tan(4 degrees) * 842 ft
Calculating this using a calculator, we get:
BC ≈ 58.73 ft
Therefore, the distance from the target B to the point C directly under the balloon is approximately 58.73 ft.