If the angle of elevation of the sun decreases from `45^(@)` to`30^(@)` , then the length of the shadow of a pillar increases by 60m . The height of the pillar is

Options

A. `60(sqrt(3)+1)`metre
B. `30(sqrt(3)-1)`metre
C. `30(sqrt(3)+1)` metre
D. `30(sqrt(3)-1)` metre

Answer

To solve the problem step by step, we can use trigonometric ratios related to the angles of elevation of the sun and the height of the pillar.

### Step 1: Understand the Problem
We have a pillar of height \( h \) meters. The angle of elevation of the sun decreases from \( 45^\circ \) to \( 30^\circ \), causing the length of the shadow to increase by \( 60 \) meters.

### Step 2: Set Up the Right Triangles
1. When the angle of elevation is \( 45^\circ \):
- Let the length of the shadow be \( x \) meters.
- Using the tangent function, we have:
\[ \tan(45^\circ) = \frac{h}{x} \]
- Since \( \tan(45^\circ) = 1 \), we can write:
\[ h = x \]

2. When the angle of elevation is \( 30^\circ \):
- The length of the shadow is now \( x + 60 \) meters.
- Using the tangent function again, we have:
\[ \tan(30^\circ) = \frac{h}{x + 60} \]
- Since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), we can write:
\[ \frac{h}{x + 60} = \frac{1}{\sqrt{3}} \quad \Rightarrow \quad h = \frac{x + 60}{\sqrt{3}} \]

### Step 3: Set Up the Equation
Now we have two equations:
1. \( h = x \) (from the first triangle)
2. \( h = \frac{x + 60}{\sqrt{3}} \) (from the second triangle)

Since both expressions equal \( h \), we can set them equal to each other:
\[ x = \frac{x + 60}{\sqrt{3}} \]

### Step 4: Solve for \( x \)
To eliminate the fraction, multiply both sides by \( \sqrt{3} \):
\[ x \sqrt{3} = x + 60 \]
Rearranging gives:
\[ x \sqrt{3} - x = 60 \]
Factoring out \( x \):
\[ x(\sqrt{3} - 1) = 60 \]
Now, solve for \( x \):
\[ x = \frac{60}{\sqrt{3} - 1} \]

### Step 5: Rationalize the Denominator
To simplify \( x \), multiply the numerator and the denominator by \( \sqrt{3} + 1 \):
\[ x = \frac{60(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{60(\sqrt{3} + 1)}{3 - 1} = \frac{60(\sqrt{3} + 1)}{2} = 30(\sqrt{3} + 1) \]

### Step 6: Find the Height \( h \)
Since \( h = x \), we have:
\[ h = 30(\sqrt{3} + 1) \]

### Final Answer
The height of the pillar is:
\[ h = 30\sqrt{3} + 30 \text{ meters} \]