If the angle of elevation of a tower from a distance of 100 m from its foot is `60^(@)`, then what will be the height of the tower?

Answer

To find the height of the tower given the angle of elevation and the distance from the tower, we can use trigonometry. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We have a right triangle formed by the height of the tower (h), the distance from the foot of the tower (100 m), and the line of sight to the top of the tower. The angle of elevation from the distance to the top of the tower is given as 60°.

### Step 2: Identify the Right Triangle
In the right triangle:
- The height of the tower (h) is the opposite side to the angle of elevation.
- The distance from the foot of the tower (100 m) is the adjacent side to the angle of elevation.
- The angle of elevation is 60°.

### Step 3: Use the Tangent Function
We can use the tangent function, which relates the opposite side (height of the tower) to the adjacent side (distance from the tower):
\[ \tan(60°) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{h}{100} \]

### Step 4: Substitute the Value of Tangent
From trigonometric tables or a calculator, we know that:
\[ \tan(60°) = \sqrt{3} \]
Now, substituting this value into the equation:
\[ \sqrt{3} = \frac{h}{100} \]

### Step 5: Solve for h
To find the height (h), we rearrange the equation:
\[ h = 100 \cdot \tan(60°) = 100 \cdot \sqrt{3} \]

### Step 6: Calculate the Height
Now we can calculate the height:
\[ h = 100 \cdot \sqrt{3} \approx 100 \cdot 1.732 \approx 173.2 \text{ m} \]

### Final Answer
The height of the tower is approximately **173.2 m**.

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