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 \(30^\circ\), we can use trigonometric ratios. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We have a tower (AB) of height 150 ft and the sun is at an elevation of \(30^\circ\). We need to find the length of the shadow (BC) cast by the tower.

### Step 2: Draw a Diagram
Draw a right triangle where:
- AB is the height of the tower (150 ft).
- BC is the length of the shadow.
- Angle A (the angle of elevation) is \(30^\circ\).

### Step 3: Identify the Right Triangle
In triangle ABC:
- AB is the opposite side (height of the tower).
- BC is the adjacent side (length of the shadow).
- Angle A is \(30^\circ\).

### Step 4: Use the Tangent Function
We can use the tangent function, which relates the opposite side to the adjacent side in a right triangle:
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \]
For our triangle:
\[ \tan(30^\circ) = \frac{AB}{BC} \]
Substituting the known values:
\[ \tan(30^\circ) = \frac{150}{BC} \]

### Step 5: Find the Value of \(\tan(30^\circ)\)
We know that:
\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \]
So we can substitute this into the equation:
\[ \frac{1}{\sqrt{3}} = \frac{150}{BC} \]

### Step 6: Solve for BC
Cross-multiply to solve for BC:
\[ BC = 150 \cdot \sqrt{3} \]

### Step 7: Final Result
Thus, the length of the shadow (BC) is:
\[ BC = 150\sqrt{3} \text{ ft} \]