If the length of the shadow of a vertical pole on the horizontal ground is 3 times its height, then the angle of elevation
A `40^@`
B. `50^@`
C. `30^@`
D. `45^@`

Options

A. C
B. D
C. A
D. B

Answer

To solve the problem, we will use trigonometric ratios to find the angle of elevation of the pole based on the given information about the shadow.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have a vertical pole and the length of its shadow is given to be 3 times the height of the pole. Let the height of the pole be \( h \) and the length of the shadow be \( s \). According to the problem, we can express this relationship as:
\[ s = 3h \]

2. **Setting Up the Triangle**:
When we draw the scenario, we form a right triangle where:
- The height of the pole \( h \) is one side (opposite side).
- The length of the shadow \( s \) is the other side (adjacent side).
- The angle of elevation \( \theta \) is the angle between the ground and the line of sight to the top of the pole.

3. **Using Trigonometric Ratios**:
We can use the tangent function, which is defined as the ratio of the opposite side to the adjacent side:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{s} \]
Substituting the values we have:
\[ \tan(\theta) = \frac{h}{3h} = \frac{1}{3} \]

4. **Finding the Angle**:
Now we need to determine the angle \( \theta \) for which \( \tan(\theta) = \frac{1}{3} \). From trigonometric tables or a calculator, we find:
\[ \theta = \tan^{-1}\left(\frac{1}{3}\right) \]
This corresponds to an angle of \( 30^\circ \).

5. **Final Answer**:
Therefore, the angle of elevation of the pole is:
\[ \theta = 30^\circ \]

### Conclusion:
The correct option is C. \( 30^\circ \).