The angle of elevation of the top of a vertical tower, from a point in the horizontal plane passing through the foot of the tower is `theta`. On moving towards the tower 192 metres, the angle of elevation becomes `phi`. If `tan theta=5/12" and "tanphi=3/4`, then find the height of the tower.

Answer

To solve the problem step by step, we will use the information provided about the angles of elevation and the distances involved.

### Step 1: Understand the problem setup
Let the height of the tower be \( h \) meters. The point from which the angle of elevation is \( \theta \) is at a distance \( x \) meters from the base of the tower. When the observer moves 192 meters closer to the tower, the new distance from the tower becomes \( x - 192 \) meters, and the angle of elevation is \( \phi \).

### Step 2: Use the tangent function for both angles
From the definitions of the tangent function, we have:
- For angle \( \theta \):
\[ \tan \theta = \frac{h}{x} \]
Given that \( \tan \theta = \frac{5}{12} \), we can write:
\[ \frac{h}{x} = \frac{5}{12} \quad \text{(1)} \]

- For angle \( \phi \):
\[ \tan \phi = \frac{h}{x - 192} \]
Given that \( \tan \phi = \frac{3}{4} \), we can write:
\[ \frac{h}{x - 192} = \frac{3}{4} \quad \text{(2)} \]

### Step 3: Express \( h \) in terms of \( x \)
From equation (1):
\[ h = \frac{5}{12} x \quad \text{(3)} \]

### Step 4: Express \( h \) in terms of \( x - 192 \)
From equation (2):
\[ h = \frac{3}{4} (x - 192) \quad \text{(4)} \]

### Step 5: Set equations (3) and (4) equal to each other
Since both (3) and (4) equal \( h \), we can set them equal:
\[ \frac{5}{12} x = \frac{3}{4} (x - 192) \]

### Step 6: Solve for \( x \)
First, clear the fractions by multiplying through by 48 (the least common multiple of 12 and 4):
\[ 48 \cdot \frac{5}{12} x = 48 \cdot \frac{3}{4} (x - 192) \]
This simplifies to:
\[ 20x = 36(x - 192) \]
Expanding the right side:
\[ 20x = 36x - 6912 \]
Rearranging gives:
\[ 6912 = 36x - 20x \]
\[ 6912 = 16x \]
\[ x = \frac{6912}{16} = 432 \text{ meters} \]

### Step 7: Substitute \( x \) back to find \( h \)
Now substitute \( x = 432 \) back into equation (3) to find \( h \):
\[ h = \frac{5}{12} \cdot 432 = 180 \text{ meters} \]

### Conclusion
The height of the tower is \( \boxed{180} \) meters.