A person standing on the bank of a river observes that the angle of elevation of the top of a tree standing on the opposite bank is `60^(@)`. When he moves 40 meter away from the bank, he finds the angle of elevation to be `30^(@)`. Find the height of the tree and width of the river
`" [take", sqrt(3)=1.732]`

Answer

To solve the problem, we will use trigonometric relationships in right triangles formed by the observer, the tree, and the river.

### Step-by-Step Solution:

1. **Define the Variables:**
- Let \( h \) be the height of the tree.
- Let \( x \) be the width of the river.

2. **Set Up the First Triangle:**
- When the person is standing at point A (on the bank), the angle of elevation to the top of the tree (point C) is \( 60^\circ \).
- The distance from point A to the base of the tree (point B) is \( x \).
- Using the tangent function:
\[ \tan(60^\circ) = \frac{h}{x} \]
- Since \( \tan(60^\circ) = \sqrt{3} \):
\[ \sqrt{3} = \frac{h}{x} \implies h = x \sqrt{3} \quad \text{(Equation 1)} \]

3. **Set Up the Second Triangle:**
- After moving 40 meters back to point D, the angle of elevation to the top of the tree is \( 30^\circ \).
- The distance from point D to the base of the tree (point B) is \( x + 40 \).
- Using the tangent function again:
\[ \tan(30^\circ) = \frac{h}{x + 40} \]
- Since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \):
\[ \frac{1}{\sqrt{3}} = \frac{h}{x + 40} \implies h = \frac{x + 40}{\sqrt{3}} \quad \text{(Equation 2)} \]

4. **Equate the Two Expressions for \( h \):**
- From Equation 1 and Equation 2:
\[ x \sqrt{3} = \frac{x + 40}{\sqrt{3}} \]
- Multiply both sides by \( \sqrt{3} \) to eliminate the fraction:
\[ 3x = x + 40 \]
- Rearranging gives:
\[ 3x - x = 40 \implies 2x = 40 \implies x = 20 \quad \text{(Width of the river)} \]

5. **Find the Height of the Tree:**
- Substitute \( x = 20 \) back into Equation 1:
\[ h = 20 \sqrt{3} \]
- Using \( \sqrt{3} \approx 1.732 \):
\[ h \approx 20 \times 1.732 = 34.64 \quad \text{(Height of the tree)} \]

### Final Answers:
- Height of the tree \( h \approx 34.64 \) meters.
- Width of the river \( x = 20 \) meters.