The angle of elevation of the top of a tower from a point on the ground which is 30 m away from the foot of the tower , is `30^(@)` . Then , the height ( in m) of the tower is .

Options

A. `(10)/(sqrt(3))`
B. `30 sqrt(3)`
C. `10 sqrt(3)`
D. `10`

Answer

To find the height of the tower based on the given information, we can follow these steps:

### Step 1: Understand the problem
We have a tower and a point on the ground that is 30 meters away from the foot of the tower. The angle of elevation from this point to the top of the tower is 30 degrees. We need to find the height of the tower.

### Step 2: Set up the right triangle
We can visualize this situation as a right triangle where:
- The height of the tower is the perpendicular side (let's denote it as \( H \)).
- The distance from the point on the ground to the foot of the tower is the base of the triangle, which is 30 m.
- The angle of elevation is 30 degrees.

### Step 3: Use the tangent function
In a right triangle, the tangent of an angle is defined as the ratio of the opposite side (height of the tower) to the adjacent side (distance from the foot of the tower). Therefore, we can write:
\[ \tan(30^\circ) = \frac{H}{30} \]

### Step 4: Solve for \( H \)
We know that:
\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \]
Substituting this value into the equation gives:
\[ \frac{1}{\sqrt{3}} = \frac{H}{30} \]
Now, we can solve for \( H \):
\[ H = 30 \times \tan(30^\circ) = 30 \times \frac{1}{\sqrt{3}} = \frac{30}{\sqrt{3}} \]

### Step 5: Rationalize the denominator
To express \( H \) in a more standard form, we can rationalize the denominator:
\[ H = \frac{30 \sqrt{3}}{3} = 10 \sqrt{3} \]

### Conclusion
Thus, the height of the tower is:
\[ H = 10\sqrt{3} \text{ meters} \]