An observer at a distance of 10 m from a tree looks at the top of the tree , the angle of elevation is `60^(@)` . What is the height of the tree ? (`sqrt(3) = 1 .73`)

Answer

To find the height of the tree using the information given, we can use the concept of trigonometry, specifically the tangent function. Here’s how to solve the problem step by step:

### Step 1: Understand the right triangle
We have a right triangle formed by:
- The height of the tree (which we want to find) as the opposite side,
- The distance from the observer to the tree as the adjacent side (10 m),
- The angle of elevation from the observer's eye to the top of the tree (60 degrees).

### Step 2: Use the tangent function
The tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side. Therefore, we can write:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
For our scenario:
\[ \tan(60^\circ) = \frac{h}{10} \]

### Step 3: Substitute the value of \(\tan(60^\circ)\)
We know from trigonometric values that:
\[ \tan(60^\circ) = \sqrt{3} \]
So we can substitute this into our equation:
\[ \sqrt{3} = \frac{h}{10} \]

### Step 4: Solve for \(h\)
To find \(h\), we can rearrange the equation:
\[ h = 10 \cdot \sqrt{3} \]

### Step 5: Substitute the value of \(\sqrt{3}\)
Given that \(\sqrt{3} \approx 1.73\), we can substitute this value:
\[ h = 10 \cdot 1.73 = 17.3 \text{ m} \]

### Conclusion
The height of the tree is approximately **17.3 meters**.

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