A man standing on the bank of a river observes that the angle of eleva...
Given:
- Angle of elevation from the man on the bank of the river to the top of the tree is 60 degrees.
- Angle of elevation from a point at a distance y m from the bank to the top of the tree is 30 degrees.
To find:
The height of the tree.
Let's assume the height of the tree is h meters.
Step 1: Finding the distance between the man and the tree
- From the given information, we can form a right-angled triangle.
- The angle of elevation from the man to the top of the tree is 60 degrees.
- Therefore, the angle between the horizontal line and the line connecting the man to the tree is 90 - 60 = 30 degrees.
- This angle is the same as the angle of elevation from the point y meters away from the bank to the top of the tree.
- So, the distance between the man and the tree is also y meters.
Step 2: Applying trigonometry
- In the right-angled triangle formed by the man, the tree, and the line connecting them, the opposite side is the height of the tree (h) and the adjacent side is the distance between the man and the tree (y).
- The ratio of the opposite side to the adjacent side is given by the tangent of the angle of elevation (30 degrees).
- Therefore, we have tan(30) = h/y.
Step 3: Solving for h
- Rearranging the equation, we get h = y * tan(30).
Step 4: Evaluating the height of the tree
- Since tan(30) = 1/sqrt(3), we can substitute this value into the equation.
- Therefore, h = y * 1/sqrt(3) = y/sqrt(3) = (y * sqrt(3))/3.
Conclusion:
The height of the tree is (y * sqrt(3))/3 meters. Therefore, the answer is option C.