The length of the shadow of a pillar is 9 m when the angle of depression of the top of the pillar is ` 60^(@)` . Find the length of the pillar

Options

A. 9
B. ` 3 sqrt""3 `
C. ` 9 sqrt"" 3 `
D. None of these

Answer

To solve the problem, we need to find the height of the pillar given the length of its shadow and the angle of depression from the top of the pillar. Here's the step-by-step solution:

### Step 1: Understand the Geometry
We have a pillar (BC) and its shadow (AB). The angle of depression from the top of the pillar (point C) to the end of the shadow (point A) is 60 degrees. The length of the shadow (AB) is given as 9 meters.

### Step 2: Identify the Angles
Since the angle of depression is 60 degrees, the angle of elevation from point A to point C is also 60 degrees (alternate interior angles).

### Step 3: Set Up the Right Triangle
In the right triangle ABC:
- AB is the base (length of the shadow) = 9 m
- BC is the height of the pillar (which we need to find, let's denote it as H)
- Angle A (angle of elevation) = 60 degrees

### Step 4: Use the Tangent Function
In triangle ABC, we can use the tangent function:
\[ \tan(\text{Angle A}) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{BC}{AB} \]
Substituting the known values:
\[ \tan(60^\circ) = \frac{H}{9} \]

### Step 5: Find the Value of Tan(60 degrees)
We know that:
\[ \tan(60^\circ) = \sqrt{3} \]
So, we can rewrite the equation:
\[ \sqrt{3} = \frac{H}{9} \]

### Step 6: Solve for H
To find H, we multiply both sides by 9:
\[ H = 9 \cdot \sqrt{3} \]

### Step 7: Final Answer
Thus, the height of the pillar is:
\[ H = 9\sqrt{3} \text{ meters} \]

### Summary
The length of the pillar is \( 9\sqrt{3} \) meters.

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