The angle of elevation of the top of a tower at a point on the ground, 50 m away from the foot of the tower, is 60°. Find 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 trigonometric ratios. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We have a right triangle formed by the height of the tower, the distance from the point on the ground to the foot of the tower, and the line of sight to the top of the tower. The angle of elevation from the point on the ground to the top of the tower is given as 60°, and the distance from the foot of the tower to the point on the ground is 50 m.

### Step 2: Identify the Right Triangle
Let:
- \( AB \) be the height of the tower (which we need to find).
- \( AC \) be the distance from the foot of the tower to the point on the ground (50 m).
- \( BC \) be the line of sight from the point on the ground to the top of the tower.

### 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:
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{AB}{AC} \]
Given that \( \theta = 60° \) and \( AC = 50 \, m \):
\[ \tan(60°) = \frac{AB}{50} \]

### Step 4: Substitute the Value of Tangent
The value of \( \tan(60°) \) is \( \sqrt{3} \):
\[ \sqrt{3} = \frac{AB}{50} \]

### Step 5: Solve for the Height of the Tower (AB)
To find \( AB \), we can rearrange the equation:
\[ AB = 50 \cdot \sqrt{3} \]

### Step 6: Calculate the Height
Now, we can calculate the height:
\[ AB = 50 \cdot \sqrt{3} \approx 50 \cdot 1.732 \approx 86.6 \, m \]

### Final Answer
The height of the tower is approximately \( 86.6 \, m \).

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