The angle of elevation of the top of a tower from a point on the ground, which is 40 m away from the foot of the tower is `30^(@)`. Find the height of the tower.

Answer

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

### Step 1: Understand the Problem
We have a tower (let's denote its height as \( h \)) and a point on the ground that is 40 m away from the base of the tower. The angle of elevation from this point to the top of the tower is \( 30^\circ \).

### Step 2: Draw a Diagram
Draw a right triangle where:
- The base (horizontal distance from the point to the tower) is 40 m.
- The height of the tower is \( h \).
- The angle of elevation \( \angle BAC = 30^\circ \).

### Step 3: Use the Tangent Function
In a right triangle, the tangent of an angle is defined as the ratio of the opposite side to the adjacent side. Here, we can use the tangent of the angle of elevation:

\[ \tan(30^\circ) = \frac{\text{Height of the tower (h)}}{\text{Distance from the tower (40 m)}} \]

### Step 4: Substitute the Known Values
We know that \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \). Therefore, we can write:

\[ \frac{1}{\sqrt{3}} = \frac{h}{40} \]

### Step 5: Solve for \( h \)
To find \( h \), we can rearrange the equation:

\[ h = 40 \cdot \tan(30^\circ) \]

Substituting the value of \( \tan(30^\circ) \):

\[ h = 40 \cdot \frac{1}{\sqrt{3}} \]

### Step 6: Rationalize the Denominator
To express \( h \) in a more standard form, we can rationalize the denominator:

\[ h = \frac{40}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{40\sqrt{3}}{3} \]

### Step 7: Calculate the Approximate Value
Using the approximate value of \( \sqrt{3} \approx 1.732 \):

\[ h \approx \frac{40 \cdot 1.732}{3} \approx \frac{69.28}{3} \approx 23.09 \text{ m} \]

### Final Answer
Thus, the height of the tower is approximately \( 23.1 \) meters.

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