The angle of elevation `theta` of the top of vertical tower from a point on the ground is `tantheta` where `tantheta=5/12`. One walking 192 meters towards the tower in the same straight line, the elevation is `alpha` where `tanalpha=3/4`. Find the height of the tower.

Answer

To find the height of the tower based on the given information, we can follow these steps:

### Step 1: Set up the problem
Let \( H \) be the height of the tower. From point A (the initial position), the angle of elevation to the top of the tower is \( \theta \) where \( \tan \theta = \frac{5}{12} \). The distance from point A to the base of the tower (point C) is \( AC = x \).

### Step 2: Use the tangent function for the first triangle
Using the tangent function for triangle ADC:
\[ \tan \theta = \frac{H}{AC} \]
Substituting the values:
\[ \frac{5}{12} = \frac{H}{x} \]
This gives us the equation:
\[ H = \frac{5}{12} x \quad \text{(Equation 1)} \]

### Step 3: Move to point B
When walking 192 meters towards the tower, the new distance from point B to the tower's base is:
\[ BC = x - 192 \]
At point B, the angle of elevation is \( \alpha \) where \( \tan \alpha = \frac{3}{4} \).

### Step 4: Use the tangent function for the second triangle
Using the tangent function for triangle BDC:
\[ \tan \alpha = \frac{H}{BC} \]
Substituting the values:
\[ \frac{3}{4} = \frac{H}{x - 192} \]
This gives us the equation:
\[ H = \frac{3}{4} (x - 192) \quad \text{(Equation 2)} \]

### Step 5: Set the two equations for height equal
Now we have two expressions for \( H \):
1. \( H = \frac{5}{12} x \)
2. \( H = \frac{3}{4} (x - 192) \)

Setting them equal to each other:
\[ \frac{5}{12} x = \frac{3}{4} (x - 192) \]

### Step 6: Solve for \( x \)
Cross-multiplying to eliminate the fractions:
\[ 5 \cdot 4x = 3 \cdot 12(x - 192) \]
\[ 20x = 36(x - 192) \]
Expanding the right side:
\[ 20x = 36x - 6912 \]
Rearranging gives:
\[ 36x - 20x = 6912 \]
\[ 16x = 6912 \]
\[ x = \frac{6912}{16} = 432 \]

### Step 7: Substitute \( x \) back to find \( H \)
Now substitute \( x \) back into Equation 1 to find \( H \):
\[ H = \frac{5}{12} \cdot 432 \]
Calculating gives:
\[ H = \frac{2160}{12} = 180 \]

### Final Answer
The height of the tower is \( H = 180 \) meters.

---