A man standing on the bank of a river observes that the angle of elevation of the top of a tree just on the opposite bank is `60^(@). `The angle of elevation is `30^(@)` from a point at a distance y m from the bank. What is the height of the tree?

Options

A. A. y m
B. B. 2y m
C. C. `( sqrt3 y)/(2) m`
D. D. `(y)/(2) m`

Answer

To solve the problem step by step, we will use the principles of trigonometry, particularly the tangent function, which relates the angles of elevation to the height of the tree and the distances involved.

### Step-by-Step Solution:

1. **Understand the Setup**:
- Let \( H \) be the height of the tree.
- Let \( X \) be the distance from the man to the base of the tree across the river.
- Let \( Y \) be the distance from the man to the bank of the river.

2. **Using the Angle of Elevation from the Bank**:
- From the bank of the river, the angle of elevation to the top of the tree is \( 60^\circ \).
- Using the tangent function:
\[ \tan(60^\circ) = \frac{H}{X} \]
- We know that \( \tan(60^\circ) = \sqrt{3} \), so:
\[ \sqrt{3} = \frac{H}{X} \]
- Rearranging gives us:
\[ H = \sqrt{3}X \quad \text{(Equation 1)} \]

3. **Using the Angle of Elevation from Distance \( Y \)**:
- From a point \( Y \) meters away from the bank, the angle of elevation to the top of the tree is \( 30^\circ \).
- Using the tangent function again:
\[ \tan(30^\circ) = \frac{H}{X + Y} \]
- We know that \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), so:
\[ \frac{1}{\sqrt{3}} = \frac{H}{X + Y} \]
- Rearranging gives us:
\[ H = \frac{1}{\sqrt{3}}(X + Y) \quad \text{(Equation 2)} \]

4. **Equating the Two Expressions for \( H \)**:
- From Equation 1 and Equation 2, we have:
\[ \sqrt{3}X = \frac{1}{\sqrt{3}}(X + Y) \]
- Multiplying through by \( \sqrt{3} \) to eliminate the fraction:
\[ 3X = X + Y \]
- Rearranging gives:
\[ 3X - X = Y \implies 2X = Y \implies X = \frac{Y}{2} \]

5. **Substituting \( X \) Back to Find \( H \)**:
- Substitute \( X = \frac{Y}{2} \) back into Equation 1:
\[ H = \sqrt{3} \left(\frac{Y}{2}\right) = \frac{\sqrt{3}Y}{2} \]

### Final Answer:
The height of the tree \( H \) is:
\[ H = \frac{\sqrt{3}}{2} Y \]