The angle of elevation of the top of a pillar of height h at a point on the ground at a distance x from the pillar is `30^(@).` On walking a distance d' towards the pillar the angle of elevation becomes `60^(@)` Then, which one of the following is correct?

Options

A. `x = d + h`
B. ` x = (3d)/(2)`
C. `x = ( 3d)/(4)`
D. `x = 2d`

Answer

To solve the problem step by step, we will use trigonometric ratios to relate the height of the pillar, the distances from the pillar, and the angles of elevation.

### Step 1: Understand the setup
We have a pillar of height \( h \). At a distance \( x \) from the base of the pillar, the angle of elevation to the top of the pillar is \( 30^\circ \). After walking a distance \( d \) towards the pillar, the angle of elevation becomes \( 60^\circ \).

### Step 2: Apply the tangent function for the first position
Using the tangent function for the angle of elevation at the first position:
\[ \tan(30^\circ) = \frac{h}{x} \]
We know that \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), so we can write:
\[ \frac{1}{\sqrt{3}} = \frac{h}{x} \]
From this, we can express \( h \) in terms of \( x \):
\[ h = \frac{x}{\sqrt{3}} \tag{1} \]

### Step 3: Apply the tangent function for the second position
Now, when the person walks \( d \) units towards the pillar, the distance from the pillar becomes \( (x - d) \). The angle of elevation is now \( 60^\circ \):
\[ \tan(60^\circ) = \frac{h}{x - d} \]
We know that \( \tan(60^\circ) = \sqrt{3} \), so we can write:
\[ \sqrt{3} = \frac{h}{x - d} \]
From this, we can express \( h \) in terms of \( x \) and \( d \):
\[ h = \sqrt{3}(x - d) \tag{2} \]

### Step 4: Set equations (1) and (2) equal to each other
Since both equations represent \( h \), we can set them equal to each other:
\[ \frac{x}{\sqrt{3}} = \sqrt{3}(x - d) \]

### Step 5: Solve for \( x \)
Multiply both sides by \( \sqrt{3} \) to eliminate the fraction:
\[ x = 3(x - d) \]
Expanding the right side:
\[ x = 3x - 3d \]
Rearranging gives:
\[ 3d = 3x - x \]
\[ 3d = 2x \]
Dividing both sides by 2:
\[ x = \frac{3d}{2} \]

### Conclusion
The relationship between \( x \) and \( d \) is given by:
\[ x = \frac{3d}{2} \]