A HOT AIR BALLOON IS 70 m ABOVE THE SEA. TWO BUOYS A AND B ARE DUE EAST OF THE BALLOON. THE ANGLE OF DEPRESSION FROM THE BALLOON TO BUOY A IS 60°. THE ANGLE OF DEPRESSION FROM THE BALLOON TO THE BUOY B IS 40°. CALCULATE DISTANCE AB.
1.
We draw 2 rt triangles with a common vertical side:
1. Draw hor. line OB.
2. Locate point "A" to the left of "B" to form AB.
3. Draw a ver. line at O and label it 70 m.
4. Draw the hyp. of each triangle
with one going to A and B.
Tan60 = 70/OA, OA = 70/Tan60.
Tan40 = 70/OB, OB = 70/Tan40.
AB = OB - OA.
To calculate the distance AB, we can use trigonometry. We know that the angle of depression is the angle between the line of sight from the observer (hot air balloon) to the object (buoy) and a line parallel to the horizontal.
Let's break down the given information and use it to calculate the distance AB step by step:
1. First, draw a diagram to visualize the scenario. Draw a horizontal line to represent the sea and mark a point above it to represent the hot air balloon.
- Label the balloon point as 'B' and note that it is 70 meters above the sea level.
- Draw lines downward from the balloon to represent the line of sight to each buoy.
- Label the point where the line intersects the sea as 'A', representing buoy A, and the other point as 'C', representing buoy B.
Here's a simple representation:
C (buoy B)
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B (balloon, 70m)
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A (buoy A)
2. According to the given information, the angle of depression from the balloon to Buoy A is 60°.
- Using trigonometry, we can calculate the height of Buoy A above the sea level. Since the angle of depression is the angle between the line of sight and a line parallel to the horizontal, the angle between the line of sight and the vertical is (90° - 60°) = 30°.
- Denote the height of Buoy A above sea level as 'h'. Then, we can use the tangent function to find 'h':
tan(30°) = h / AB
h = AB * tan(30°)
h = AB * √3 / 3 (approximating √3 / 3 to 0.58)
3. Similarly, the angle of depression from the balloon to Buoy B is given as 40°.
- Using the same steps as above, denote the height of Buoy B above sea level as 'x':
tan(50°) = x / AB
x = AB * tan(50°)
4. Now, we have two equations for the height of Buoy A and Buoy B:
h = AB * 0.58
x = AB * tan(50°)
5. Since Buoy A is at sea level (height = 0), we can write the following equation using the information from steps 4 and 5:
x + h = 70
Substituting the equations for h and x from steps 4 and 5:
AB * 0.58 + AB * tan(50°) = 70
6. Now, we can solve the equation to find the value of AB:
AB * 0.58 + AB * tan(50°) = 70
AB(0.58 + tan(50°)) = 70
AB = 70 / (0.58 + tan(50°))
Using a calculator we can find that AB ≈ 45.24 meters (approx.)
Therefore, the distance AB between Buoy A and Buoy B is approximately 45.24 meters.