If the angle of elevation of sun is `theta` and the length of the shadow of a pole of length p is s, then

Options

A. `p = s cos theta`
B. `p = s sin theta`
C. ` p = (s)/( cot theta)`
D. ` p = s cot theta `

Answer

To solve the problem, we need to analyze the situation involving a pole and its shadow in relation to the angle of elevation of the sun.

### Step-by-Step Solution:

1. **Understand the Geometry**:
- We have a pole of height \( p \) and a shadow of length \( s \).
- The angle of elevation of the sun is \( \theta \).
- We can visualize this as a right triangle where:
- The height of the pole is one side (opposite to angle \( \theta \)).
- The length of the shadow is the other side (adjacent to angle \( \theta \)).

2. **Label the Triangle**:
- Let's label the triangle formed by the pole and its shadow:
- Let \( A \) be the top of the pole.
- Let \( B \) be the base of the pole (where it touches the ground).
- Let \( C \) be the tip of the shadow.
- Thus, \( AB = p \) (height of the pole), \( BC = s \) (length of the shadow), and \( AC \) is the hypotenuse.

3. **Use Trigonometric Ratios**:
- In right triangle \( ABC \), we can use the cotangent function, which relates the adjacent side to the opposite side:
\[ \cot(\theta) = \frac{\text{Adjacent}}{\text{Opposite}} = \frac{BC}{AB} \]
- Substituting the lengths:
\[ \cot(\theta) = \frac{s}{p} \]

4. **Rearranging the Equation**:
- We can rearrange the equation to express \( p \) in terms of \( s \) and \( \theta \):
\[ p = \frac{s}{\cot(\theta)} \]

5. **Final Result**:
- Therefore, the relationship between the height of the pole \( p \), the length of the shadow \( s \), and the angle of elevation \( \theta \) is given by:
\[ p = \frac{s}{\cot(\theta)} \]

### Conclusion:
The final answer to the problem is:
\[ p = \frac{s}{\cot(\theta)} \]