The shadow of a tree is `(1)/(sqrt(3))` times the length of the tree . Find the angle of elevation.

Options

A. `30^(@)`
B. `90^(@)`
C. `45^(@)`
D. `60^(@)`

Answer

To solve the problem of finding the angle of elevation given that the shadow of a tree is \( \frac{1}{\sqrt{3}} \) times the length of the tree, we can follow these steps:

### Step 1: Define the Variables
Let:
- \( AB \) be the height of the tree.
- \( BC \) be the length of the shadow.

According to the problem, we have:
\[ BC = \frac{1}{\sqrt{3}} \times AB \]
If we denote the height of the tree \( AB \) as \( x \), then the length of the shadow \( BC \) can be expressed as:
\[ BC = \frac{x}{\sqrt{3}} \]

### Step 2: Set Up the Right Triangle
In this scenario, we can visualize a right triangle where:
- The height of the tree \( AB \) is the opposite side.
- The length of the shadow \( BC \) is the adjacent side.
- The angle of elevation \( \theta \) is the angle between the ground and the line from the top of the tree to the tip of the shadow.

### Step 3: Use the Tangent Function
The tangent of the angle of elevation \( \theta \) can be expressed as:
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{AB}{BC} \]
Substituting the values we have:
\[ \tan(\theta) = \frac{x}{\frac{x}{\sqrt{3}}} \]
This simplifies to:
\[ \tan(\theta) = \sqrt{3} \]

### Step 4: Find the Angle
To find the angle \( \theta \), we need to determine the angle whose tangent is \( \sqrt{3} \). From trigonometric values, we know:
\[ \tan(60^\circ) = \sqrt{3} \]
Thus, we conclude:
\[ \theta = 60^\circ \]

### Final Answer
The angle of elevation is \( 60^\circ \).

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