A vertical tower stands on a horizontal plane, and from a point on the ground at a distance of 30 metres from the foot of the tower , the angle of elevation is `60^@`. The height of the tower is

Answer

To find the height of the vertical tower, we can use the concept of trigonometry, specifically the tangent function. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We have a vertical tower (let's denote the top of the tower as point A and the foot of the tower as point B). A point C is located on the ground at a distance of 30 meters from point B. The angle of elevation from point C to point A is given as 60 degrees.

### Step 2: Identify the Right Triangle
In triangle ABC:
- AB is the height of the tower (which we need to find).
- BC is the distance from point C to point B, which is 30 meters.
- Angle ACB is the angle of elevation, which is 60 degrees.

### 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. Therefore, we can write:

\[ \tan(\text{angle ACB}) = \frac{\text{Opposite (AB)}}{\text{Adjacent (BC)}} \]

Substituting the known values:

\[ \tan(60^\circ) = \frac{AB}{BC} \]

### Step 4: Substitute the Values
We know that:
- \(\tan(60^\circ) = \sqrt{3}\)
- \(BC = 30\) meters

So, we can substitute these values into the equation:

\[ \sqrt{3} = \frac{AB}{30} \]

### Step 5: Solve for AB
To find AB (the height of the tower), we can rearrange the equation:

\[ AB = 30 \cdot \sqrt{3} \]

### Step 6: Final Answer
Thus, the height of the tower is:

\[ AB = 30\sqrt{3} \text{ meters} \]