Find the height of a rock mountain if on moving 100m towards the rock in the horizontal direction through the base of the rock, the angle of elevation of its top increases from 30 degree to 45 degree?

Question

Answer

Answer

Answer

Given:
- The angle of elevation of the top of a rock mountain increases from 30 degrees to 45 degrees when moving 100m horizontally towards the rock through the base.

To find:
- The height of the rock mountain.

Approach:
1. We can use the tangent function to solve this problem.
2. Let's assume the height of the rock mountain is 'h' meters.
3. We can form a right-angled triangle with the base of the triangle as the horizontal distance moved towards the rock (100m) and the height of the triangle as 'h'.
4. The angle of elevation of the top of the rock mountain is the angle between the horizontal line and the line of sight to the top of the mountain.
5. In the initial position, the angle of elevation is 30 degrees, and in the final position, it is 45 degrees.
6. We can write the following equations based on the given information:
- tan(30) = h / 100
- tan(45) = h / (100 - x), where 'x' is the horizontal distance from the initial position to the final position.
7. Rearranging the equations, we get:
- h = 100 * tan(30)
- h = (100 - x) * tan(45)
8. Equating the two expressions for 'h', we can solve for 'x' and then find the value of 'h'.

Solution:
- Using a scientific calculator, we can find that tan(30) is approximately 0.5774 and tan(45) is 1.
- Substituting these values in the equations, we have:
- h = 100 * 0.5774
- h = (100 - x) * 1
- Simplifying the equations, we get:
- h = 57.74
- h = 100 - x
- Solving for 'x', we find:
- x = 100 - h
- x = 100 - 57.74
- x = 42.26
- Therefore, the height of the rock mountain is approximately 57.74 meters and the horizontal distance from the initial position to the final position is approximately 42.26 meters.

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