The angle of elevation of the top of a tower from a point 40 m away from its foot is `60^(@)`. 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 and a point from which the angle of elevation to the top of the tower is given. We need to find the height of the tower.

### Step 2: Draw a diagram
Let's denote:
- The height of the tower as \( h \).
- The distance from the point to the foot of the tower as \( d = 40 \) m.
- The angle of elevation as \( \theta = 60^\circ \).

### Step 3: Identify the right triangle
From the point at a distance of 40 m from the foot of the tower, we can form a right triangle:
- The height of the tower \( h \) is the opposite side.
- The distance from the point to the foot of the tower (40 m) is the adjacent side.
- The angle of elevation \( \theta \) is at the point.

### Step 4: Use the tangent function
In a right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side. Therefore, we can write:
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{h}{d} \]
Substituting the known values:
\[ \tan(60^\circ) = \frac{h}{40} \]

### Step 5: Calculate \( \tan(60^\circ) \)
We know that:
\[ \tan(60^\circ) = \sqrt{3} \]
So we can substitute this into our equation:
\[ \sqrt{3} = \frac{h}{40} \]

### Step 6: Solve for \( h \)
To find the height \( h \), we can rearrange the equation:
\[ h = 40 \cdot \sqrt{3} \]

### Step 7: Calculate the numerical value
Using the approximate value of \( \sqrt{3} \approx 1.732 \):
\[ h \approx 40 \cdot 1.732 \approx 69.28 \text{ m} \]

### Final Answer
The height of the tower is approximately \( 69.28 \) meters.

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