A person standing on the ground observes the angle of elevation of the top of a tower to be `30^@` On walking a distance a in a certain direction, he finds the elevation of the top to be same as before. He then walks a distance `5/3 a` at right angles to his former direction, and finds that the elevation of the top has doubled. The height of the tower is

Options

A. `sqrt(5/12)`a
B. `sqrt(5/(48))`a
C. `sqrt(6/5)a`
D. `sqrt(48/85)`a

Answer

To solve the problem step by step, we can follow these steps:

### Step 1: Understand the Problem
We have a tower of height \( h \). A person observes the top of the tower from point \( P \) at an angle of elevation of \( 30^\circ \). After walking a distance \( a \), the angle of elevation remains \( 30^\circ \). Then, after walking \( \frac{5}{3}a \) at right angles to his previous direction, the angle of elevation becomes \( 60^\circ \).

### Step 2: Draw the Diagram
Let's denote:
- \( A \) as the base of the tower,
- \( B \) as the top of the tower,
- \( P \) as the initial position of the observer,
- \( Q \) as the position after walking distance \( a \),
- \( S \) as the position after walking \( \frac{5}{3}a \) at right angles.

### Step 3: Set Up the Relationships
1. From point \( P \):
\[ \tan(30^\circ) = \frac{h}{x} \quad \text{(where \( x \) is the horizontal distance from \( P \) to \( A \))} \]
Since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \):
\[ \frac{1}{\sqrt{3}} = \frac{h}{x} \implies x = \sqrt{3}h \]

2. From point \( Q \) (after walking \( a \)):
The distance from \( Q \) to \( A \) is still \( x \) (since the angle remains \( 30^\circ \)):
\[ \tan(30^\circ) = \frac{h}{x} \implies x = \sqrt{3}h \]

### Step 4: Analyze the Position \( S \)
From point \( S \) (after walking \( \frac{5}{3}a \)):
- The distance from \( S \) to \( A \) is:
\[ SB = x - \frac{5}{3}a \]
- The angle of elevation is \( 60^\circ \):
\[ \tan(60^\circ) = \frac{h}{SB} \implies \sqrt{3} = \frac{h}{x - \frac{5}{3}a} \]
Rearranging gives:
\[ h = \sqrt{3} \left( x - \frac{5}{3}a \right) \]

### Step 5: Substitute for \( x \)
Substituting \( x = \sqrt{3}h \) into the equation:
\[ h = \sqrt{3} \left( \sqrt{3}h - \frac{5}{3}a \right) \]
Expanding gives:
\[ h = 3h - \frac{5\sqrt{3}}{3}a \]
Rearranging:
\[ 3h - h = \frac{5\sqrt{3}}{3}a \implies 2h = \frac{5\sqrt{3}}{3}a \]
Thus:
\[ h = \frac{5\sqrt{3}}{6}a \]

### Final Result
The height of the tower is:
\[ h = \frac{5\sqrt{3}}{6}a \]