Two posts are k metres apart. If from the middle point of the line joining their feet, an observer finds the angles of elevations of their tops to be `60^(@) and 30^(@)` respectively, then the ratio of heiht of the posts respectively is

Options

A. 3
B. `1/3`
C. `sqrt3`
D. `1/sqrt3`

Answer

To find the ratio of the heights of the two posts, we can follow these steps:

### Step 1: Understand the Setup
Let the heights of the two posts be \( P \) and \( A \). The distance between the two posts is \( K \) meters. The observer is standing at the midpoint \( O \) of the line joining the feet of the posts. Thus, the distance from \( O \) to each post is \( \frac{K}{2} \).

### Step 2: Set Up the Triangle for the First Post
For the first post (height \( P \)), the angle of elevation from point \( O \) is \( 60^\circ \). We can use the tangent function to relate the height of the post to the distance from the observer:

\[ \tan(60^\circ) = \frac{P}{\frac{K}{2}} \]

### Step 3: Solve for Height \( P \)
Using the value of \( \tan(60^\circ) = \sqrt{3} \):

\[ \sqrt{3} = \frac{P}{\frac{K}{2}} \]

Multiplying both sides by \( \frac{K}{2} \):

\[ P = \sqrt{3} \cdot \frac{K}{2} = \frac{\sqrt{3}K}{2} \]

### Step 4: Set Up the Triangle for the Second Post
For the second post (height \( A \)), the angle of elevation from point \( O \) is \( 30^\circ \). Again, we use the tangent function:

\[ \tan(30^\circ) = \frac{A}{\frac{K}{2}} \]

### Step 5: Solve for Height \( A \)
Using the value of \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \):

\[ \frac{1}{\sqrt{3}} = \frac{A}{\frac{K}{2}} \]

Multiplying both sides by \( \frac{K}{2} \):

\[ A = \frac{K}{2\sqrt{3}} \]

### Step 6: Find the Ratio of Heights
Now, we need to find the ratio of the heights of the two posts, \( \frac{P}{A} \):

\[ \frac{P}{A} = \frac{\frac{\sqrt{3}K}{2}}{\frac{K}{2\sqrt{3}}} \]

### Step 7: Simplify the Ratio
Cancelling \( \frac{K}{2} \) from the numerator and denominator:

\[ \frac{P}{A} = \frac{\sqrt{3}}{\frac{1}{\sqrt{3}}} = \sqrt{3} \cdot \sqrt{3} = 3 \]

### Conclusion
Thus, the ratio of the heights of the posts \( P \) and \( A \) is:

\[ \frac{P}{A} = 3 \]