Find the angle of elevation of the Sun when the shadow of a pole ` h` `m` high is `sqrt3 h` `m` long.

Options

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

Explanation

To find the angle of elevation of the Sun when the shadow of a pole \( h \) meters high is \( \sqrt{3}h \) meters long, we can follow these steps:

### Step 1: Draw the Diagram
Draw a right triangle where:
- The pole represents the vertical side (height \( h \)).
- The shadow represents the horizontal side (length \( \sqrt{3}h \)).
- The angle of elevation \( \theta \) is the angle formed between the ground and the line from the top of the pole to the tip of the shadow.

### Step 2: Identify the Sides of the Triangle
In the right triangle:
- The opposite side (height of the pole) is \( h \).
- The adjacent side (length of the shadow) is \( \sqrt{3}h \).

### Step 3: Use the Tangent Function
The tangent of the angle \( \theta \) is given by the ratio of the opposite side to the adjacent side:
\[ \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{h}{\sqrt{3}h} \]

### Step 4: Simplify the Expression
Since \( h \) is common in both the numerator and the denominator, we can cancel it out:
\[ \tan \theta = \frac{1}{\sqrt{3}} \]

### Step 5: Determine the Angle
We know from trigonometric values that:
\[ \tan 30^\circ = \frac{1}{\sqrt{3}} \]
Thus, we can conclude that:
\[ \theta = 30^\circ \]

### Final Answer
The angle of elevation of the Sun is \( 30^\circ \).

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