Find the height of a tree it is found that on walking away from is 20 m, in a horizontal line through is base , the elevation of its top changes from `60^(@) t o 30^(@)`

Answer

To find the height of the tree, we can use trigonometric principles based on the angles of elevation and the distances involved. Let's break down the solution step by step.

### Step 1: Understanding the Problem
We have a tree, and we are given two angles of elevation to the top of the tree from two different points. The first angle of elevation is \(60^\circ\) when we are at point C (closer to the tree), and the second angle of elevation is \(30^\circ\) when we are at point D (20 meters away from point C).

### Step 2: Setting Up the Diagram
Let:
- \(AB\) = height of the tree (unknown)
- \(BC\) = distance from point B (base of the tree) to point C
- \(BD\) = distance from point B to point D, which is \(BC + 20\)

### Step 3: Using Trigonometric Ratios
From triangle \(ABC\) (where angle \(ACB = 60^\circ\)):
\[ \tan(60^\circ) = \frac{AB}{BC} \]
We know that \(\tan(60^\circ) = \sqrt{3}\), so:
\[ \sqrt{3} = \frac{AB}{BC} \quad \Rightarrow \quad AB = BC \cdot \sqrt{3} \quad \text{(Equation 1)} \]

From triangle \(ABD\) (where angle \(ADB = 30^\circ\)):
\[ \tan(30^\circ) = \frac{AB}{BD} \]
We know that \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\), so:
\[ \frac{1}{\sqrt{3}} = \frac{AB}{BD} \quad \Rightarrow \quad AB = BD \cdot \frac{1}{\sqrt{3}} \]
Since \(BD = BC + 20\), we can substitute:
\[ AB = (BC + 20) \cdot \frac{1}{\sqrt{3}} \quad \text{(Equation 2)} \]

### Step 4: Equating the Two Expressions for \(AB\)
From Equation 1 and Equation 2, we have:
\[ BC \cdot \sqrt{3} = (BC + 20) \cdot \frac{1}{\sqrt{3}} \]

### Step 5: Solving for \(BC\)
Multiplying both sides by \(\sqrt{3}\):
\[ 3BC = BC + 20 \]
Rearranging gives:
\[ 3BC - BC = 20 \quad \Rightarrow \quad 2BC = 20 \quad \Rightarrow \quad BC = 10 \text{ m} \]

### Step 6: Finding the Height \(AB\)
Now substituting \(BC\) back into Equation 1:
\[ AB = BC \cdot \sqrt{3} = 10 \cdot \sqrt{3} \]
Calculating \(AB\):
\[ AB = 10 \cdot 1.732 \approx 17.32 \text{ m} \]

### Conclusion
The height of the tree is approximately \(17.32\) meters.

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