The length of the shadow of a vertical pole is `1/sqrt3` times its height. Find the angle of elevation .

Options

A. `60^0`
B. `45^0`
C. `90^0`
D. `30^0`

Explanation

To solve the problem step by step, we will follow these instructions:

### Step 1: Understand the Problem
We have a vertical pole and the length of its shadow is given as \( \frac{1}{\sqrt{3}} \) times its height. We need to find the angle of elevation of the sun, which is the angle formed between the line from the top of the pole to the tip of the shadow and the ground.

### Step 2: Define Variables
Let the height of the vertical pole be \( h \). According to the problem, the length of the shadow \( L \) can be expressed as:
\[ L = \frac{h}{\sqrt{3}} \]

### Step 3: Draw a Diagram
We can visualize the situation as a right triangle where:
- The height of the pole \( h \) is the opposite side.
- The length of the shadow \( L \) is the adjacent side.
- The angle of elevation \( \theta \) is the angle between the ground and the line from the top of the pole to the tip of the shadow.

### Step 4: Use Trigonometric Ratios
In a right triangle, the tangent of the angle \( \theta \) is given by the ratio of the opposite side to the adjacent side:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{L} \]
Substituting the expression for \( L \):
\[ \tan(\theta) = \frac{h}{\frac{h}{\sqrt{3}}} \]

### Step 5: Simplify the Expression
Now, simplify the equation:
\[ \tan(\theta) = \frac{h \cdot \sqrt{3}}{h} \]
The \( h \) in the numerator and denominator cancels out:
\[ \tan(\theta) = \sqrt{3} \]

### Step 6: Find the Angle
Now we need to find the angle \( \theta \) such that:
\[ \tan(\theta) = \sqrt{3} \]
From trigonometric values, we know that:
\[ \tan(60^\circ) = \sqrt{3} \]
Thus, we can conclude:
\[ \theta = 60^\circ \]

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

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