A tower subtends an angle of `60^(@)` at a point on the plane passing through its foot and at a point 20 m vertically above the first point, the angle of depression of the foot of tower is `45^(@)`. Find the height of the tower.

Answer

To solve the problem, we will first visualize the situation with a diagram and then apply trigonometric principles to find the height of the tower.

### Step 1: Draw the Diagram
1. **Draw a vertical tower** and label its height as \( h \).
2. **Mark point A** on the ground at the foot of the tower.
3. **Mark point B** at a point on the ground where the angle of elevation to the top of the tower is \( 60^\circ \).
4. **Mark point C** which is 20 meters vertically above point B. The angle of depression from point C to point A is \( 45^\circ \).

### Step 2: Identify the Components
- Let the height of the tower be \( h \).
- The distance from point A (foot of the tower) to point B (on the ground) will be denoted as \( x \).
- The height from point B to point C is \( 20 \) m.

### Step 3: Use Trigonometry for Triangle ABC
In triangle ABC (where C is above B):
- The angle of depression from C to A is \( 45^\circ \).
- Therefore, the angle of elevation from A to C is also \( 45^\circ \).

Using the tangent function:
\[ \tan(45^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{20}{x} \]
Since \( \tan(45^\circ) = 1 \):
\[ 1 = \frac{20}{x} \implies x = 20 \text{ m} \]

### Step 4: Use Trigonometry for Triangle ABD
In triangle ABD (where D is the top of the tower):
- The angle of elevation from point B to the top of the tower is \( 60^\circ \).

Using the tangent function again:
\[ \tan(60^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h - 20}{x} \]
Since \( \tan(60^\circ) = \sqrt{3} \):
\[ \sqrt{3} = \frac{h - 20}{20} \]

### Step 5: Solve for the Height \( h \)
Cross-multiplying gives:
\[ h - 20 = 20\sqrt{3} \]
Adding 20 to both sides:
\[ h = 20 + 20\sqrt{3} \]

### Step 6: Calculate the Height
Now, we can calculate the numerical value of \( h \):
\[ h = 20 + 20\sqrt{3} \approx 20 + 20 \times 1.732 \approx 20 + 34.64 \approx 54.64 \text{ m} \]

### Final Answer
The height of the tower is approximately \( 54.64 \) meters.