The angle of elevation `theta` of the top of a tower is `30^(@)`. If the height of the tower is doubled, then new `tan theta` will be

Options

A. `sqrt 3/2`
B. `3/2`
C. `2/3`
D. `2/sqrt 3`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the problem
We are given that the angle of elevation \( \theta \) of the top of a tower is \( 30^\circ \). We need to find the new value of \( \tan \theta \) when the height of the tower is doubled.

### Step 2: Recall the definition of tangent
The tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side. In this case, if \( h \) is the height of the tower and \( x \) is the distance from the base of the tower to the point of observation, we have:
\[ \tan(30^\circ) = \frac{h}{x} \]

### Step 3: Calculate \( \tan(30^\circ) \)
From trigonometric values, we know:
\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \]
Thus, we can write:
\[ \frac{h}{x} = \frac{1}{\sqrt{3}} \]
This implies:
\[ h = \frac{x}{\sqrt{3}} \]

### Step 4: Double the height of the tower
If the height of the tower is doubled, the new height \( h' \) will be:
\[ h' = 2h = 2 \left(\frac{x}{\sqrt{3}}\right) = \frac{2x}{\sqrt{3}} \]

### Step 5: Find the new tangent value
Now, we need to find the new value of \( \tan \theta \) with the new height \( h' \):
\[ \tan(\theta) = \frac{h'}{x} = \frac{\frac{2x}{\sqrt{3}}}{x} = \frac{2}{\sqrt{3}} \]

### Step 6: Conclusion
The new value of \( \tan \theta \) when the height of the tower is doubled is:
\[ \tan(\theta) = \frac{2}{\sqrt{3}} \]