If from a point 100m above the ground the angles of depression of two objects due south on ground are 60 and 45 , then the distance between objects is?

Question

Answer

Answer

Answer

Problem: Find the distance between two objects on the ground given the angles of depression from a point 100m above the ground.

Given:
- Angles of depression of two objects due south on ground are 60 and 45.
- The point of observation is 100m above the ground.

Solution:
We can use trigonometry to solve the problem. Let's assume that the two objects are A and B and the point of observation is C.
1. Draw a diagram: It is important to draw a diagram to understand the problem. Below is a rough diagram of the problem:

![image.png](attachment:image.png)

2. Determine the height of the objects: Since we know the angle of depression and the height of the point of observation, we can use trigonometry to find the height of the objects. Let hA and hB be the heights of objects A and B respectively. Then,

hA = 100/tan(60) ≈ 57.7m

hB = 100/tan(45) ≈ 100m

3. Find the distance between the objects: Let d be the distance between objects A and B. Then, we can use the right-angled triangle formed by objects A, B and the point of observation C to find d.

Using object A, we have: tan(60) = hA/d

Therefore, d = hA/tan(60) ≈ 99.0m

Using object B, we have: tan(45) = hB/(d + x)

where x is the distance between object A and the point of observation C.

Rearranging the equation, we get: x = hB/tan(45) - d ≈ 28.3m

4. Final answer: The distance between objects A and B is approximately 99.0m and the distance between object A and the point of observation C is approximately 28.3m. Therefore, the distance between objects A and B is approximately 99.0 + 28.3 = 127.3m.

Answer: The distance between objects A and B is approximately 127.3m.

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