From a point on the ground, the angle of elevation of the top of a vertical tower is found to be such that its tangent is `(3)/(5)` . On walking 50 m towards the tower, the tangent of the new angle of elevation of the top of the tower is found to be `(4)/(5)` . 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: Set up the problem
Let the height of the tower be \( h \) meters. From the first point on the ground, the tangent of the angle of elevation to the top of the tower is given as \( \tan(\alpha) = \frac{3}{5} \).

### Step 2: Use the tangent definition
From the definition of tangent in triangle \( ABC \):
\[ \tan(\alpha) = \frac{h}{BC} \]
where \( BC \) is the horizontal distance from the first point to the base of the tower. Thus, we can write:
\[ \frac{3}{5} = \frac{h}{BC} \implies h = \frac{3}{5} BC \tag{1} \]

### Step 3: Move 50 meters closer to the tower
After walking 50 meters towards the tower, the new distance from the tower becomes \( BC - 50 \). The tangent of the new angle of elevation \( \beta \) is given as \( \tan(\beta) = \frac{4}{5} \).

### Step 4: Set up the second tangent equation
Using the tangent definition in triangle \( ABD \):
\[ \tan(\beta) = \frac{h}{BD} \]
where \( BD = BC - 50 \). Thus, we can write:
\[ \frac{4}{5} = \frac{h}{BC - 50} \implies h = \frac{4}{5}(BC - 50) \tag{2} \]

### Step 5: Equate the two expressions for height
Now we have two expressions for \( h \) from equations (1) and (2):
\[ \frac{3}{5} BC = \frac{4}{5}(BC - 50) \]

### Step 6: Eliminate the fractions
To eliminate the fractions, multiply the entire equation by 5:
\[ 3BC = 4(BC - 50) \]
Expanding the right side gives:
\[ 3BC = 4BC - 200 \]

### Step 7: Solve for \( BC \)
Rearranging the equation:
\[ 3BC - 4BC = -200 \implies -BC = -200 \implies BC = 200 \text{ meters} \]

### Step 8: Substitute back to find \( h \)
Now substitute \( BC = 200 \) back into equation (1) to find \( h \):
\[ h = \frac{3}{5} \times 200 = 120 \text{ meters} \]

### Final Answer
The height of the tower is \( \boxed{120} \) meters.

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