If the elevation of the sun is `30^@` , then the length of the shadow cast by a tower of 150 ft. height is

Options

A. `75sqrt3` ft.
B. `200sqrt3` ft
C. `150sqrt3` ft.
D. none of these

Answer

To find the length of the shadow cast by a tower of height 150 ft when the elevation of the sun is 30 degrees, we can use the concept of right triangles and trigonometric ratios.

### Step-by-Step Solution:

1. **Understand the Problem**: We have a tower of height \( h = 150 \) ft. The angle of elevation of the sun is \( \theta = 30^\circ \). We need to find the length of the shadow \( S \).

2. **Draw a Diagram**: Visualize a right triangle where:
- The height of the tower is the vertical side (150 ft).
- The length of the shadow is the horizontal side (S).
- The angle of elevation from the tip of the shadow to the top of the tower is \( 30^\circ \).

3. **Use the Tangent Function**: The tangent of the angle of elevation is defined as the ratio of the opposite side (height of the tower) to the adjacent side (length of the shadow):
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{h}{S} \]
Substituting the known values:
\[ \tan(30^\circ) = \frac{150}{S} \]

4. **Calculate \( \tan(30^\circ) \)**: We know from trigonometric tables that:
\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \approx 0.577 \]

5. **Set Up the Equation**: Now substituting this value into the equation:
\[ \frac{1}{\sqrt{3}} = \frac{150}{S} \]

6. **Cross-Multiply to Solve for \( S \)**:
\[ S = 150 \cdot \sqrt{3} \]

7. **Calculate the Length of the Shadow**: Using the approximate value of \( \sqrt{3} \approx 1.732 \):
\[ S \approx 150 \cdot 1.732 \approx 259.8 \text{ ft} \]

### Final Answer:
The length of the shadow cast by the tower is approximately \( 259.8 \) ft.

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