A vertical tower stands on a horizontal plane and is surmounted by a vertical flag staff of height 6 meters. At point on the plane, the angle of elvation of the bottom and the top of the flag are respectively `30^@ and 60^@`. Find the height of tower

Answer

To solve the problem, we will break it down step by step:

### Step 1: Understand the problem and draw a diagram
We have a vertical tower with a height we need to find, and it has a flagstaff of height 6 meters on top of it. The angles of elevation from a point on the ground to the bottom of the tower and the top of the flagstaff are given as 30° and 60°, respectively.

### Step 2: Set up the diagram
Let's label:
- Point A: The point on the ground where the observer is standing.
- Point B: The bottom of the tower.
- Point C: The top of the tower.
- Point D: The top of the flagstaff.

Let the height of the tower be \( H \) meters. Therefore, the height from the ground to the top of the flagstaff (point D) will be \( H + 6 \) meters.

### Step 3: Use the tangent function for angle 60°
From point A, the angle of elevation to point D (top of the flagstaff) is 60°.
Using the tangent function:
\[ \tan(60°) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{H + 6}{d} \]
Where \( d \) is the horizontal distance from point A to point B (the base of the tower).
Since \( \tan(60°) = \sqrt{3} \):
\[ \sqrt{3} = \frac{H + 6}{d} \quad \text{(1)} \]

### Step 4: Use the tangent function for angle 30°
From point A, the angle of elevation to point B (bottom of the tower) is 30°.
Using the tangent function:
\[ \tan(30°) = \frac{H}{d} \]
Since \( \tan(30°) = \frac{1}{\sqrt{3}} \):
\[ \frac{1}{\sqrt{3}} = \frac{H}{d} \quad \text{(2)} \]

### Step 5: Solve for \( d \) in both equations
From equation (1):
\[ d = \frac{H + 6}{\sqrt{3}} \quad \text{(3)} \]
From equation (2):
\[ d = H \sqrt{3} \quad \text{(4)} \]

### Step 6: Set equations (3) and (4) equal to each other
Since both equations represent \( d \):
\[ \frac{H + 6}{\sqrt{3}} = H \sqrt{3} \]

### Step 7: Cross-multiply and simplify
Cross-multiplying gives:
\[ H + 6 = 3H \]
Rearranging this gives:
\[ 3H - H = 6 \]
\[ 2H = 6 \]
\[ H = 3 \]

### Conclusion
The height of the tower \( H \) is 3 meters.