At a point 20 m away from the foot of a tower, the angle of elevation of the top of the tower is `30^@` The height of the tower is

Options

A. `20 sqrt(3) m`
B. `(20)/(sqrt(3))m`
C. `(sqrt(3))/(20m )`
D. None of these

Answer

To find the height of the tower based on the given information, we can use trigonometric relationships. Here’s a step-by-step solution:

### Step 1: Understand the scenario
We have a tower, and we are standing 20 meters away from its base. The angle of elevation to the top of the tower is 30 degrees.

### Step 2: Define the variables
Let:
- \( BC \) = height of the tower (which we need to find)
- \( AB \) = distance from the point to the foot of the tower = 20 m
- \( \angle A = 30^\circ \)

### Step 3: Use the tangent function
From trigonometry, we know that:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
In our case:
\[ \tan(30^\circ) = \frac{BC}{AB} \]
Substituting the known values:
\[ \tan(30^\circ) = \frac{BC}{20} \]

### Step 4: Calculate \( \tan(30^\circ) \)
We know that:
\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \]
So we can rewrite our equation:
\[ \frac{1}{\sqrt{3}} = \frac{BC}{20} \]

### Step 5: Solve for \( BC \)
Now, we can cross-multiply to solve for \( BC \):
\[ BC = 20 \cdot \frac{1}{\sqrt{3}} \]
This simplifies to:
\[ BC = \frac{20}{\sqrt{3}} \text{ meters} \]

### Step 6: Rationalize the denominator (optional)
To express this in a more standard form, we can rationalize the denominator:
\[ BC = \frac{20 \sqrt{3}}{3} \text{ meters} \]

### Final Answer
Thus, the height of the tower is:
\[ \frac{20 \sqrt{3}}{3} \text{ meters} \]

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