At a point, the angle of elevation of a tower is such that its tangent is `5/12`. On walking 240 m nearer to the tower, the tangent of the angle of elevation becomes `3/4`. 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: Understanding the Problem
We have a tower and two points of observation. The tangent of the angle of elevation from the first point is \( \frac{5}{12} \) and from the second point (after walking 240 m closer) is \( \frac{3}{4} \).

### Step 2: Setting Up the Variables
Let:
- \( h \) = height of the tower
- \( x \) = distance from the first point to the base of the tower
- \( y \) = distance from the second point to the base of the tower

From the problem, we know:
- \( y = x - 240 \) (since we walked 240 m closer to the tower)

### Step 3: Using the Tangent Ratios
From the first point:
\[ \tan(\theta) = \frac{h}{x} = \frac{5}{12} \]
This gives us:
\[ h = \frac{5}{12} x \quad \text{(Equation 1)} \]

From the second point:
\[ \tan(\theta') = \frac{h}{y} = \frac{3}{4} \]
This gives us:
\[ h = \frac{3}{4} y \quad \text{(Equation 2)} \]

### Step 4: Substitute \( y \) in Equation 2
Substituting \( y = x - 240 \) into Equation 2:
\[ h = \frac{3}{4}(x - 240) \]
Now we have:
\[ h = \frac{3}{4}x - 180 \quad \text{(Equation 3)} \]

### Step 5: Equating the Two Expressions for \( h \)
From Equation 1 and Equation 3, we can set them equal to each other:
\[ \frac{5}{12} x = \frac{3}{4} x - 180 \]

### Step 6: Solving for \( x \)
To eliminate the fractions, we can multiply the entire equation by 12:
\[ 5x = 9x - 2160 \]
Rearranging gives:
\[ 2160 = 9x - 5x \]
\[ 2160 = 4x \]
\[ x = \frac{2160}{4} = 540 \text{ m} \]

### Step 7: Finding the Height \( h \)
Now substitute \( x = 540 \) back into Equation 1 to find \( h \):
\[ h = \frac{5}{12} \times 540 = \frac{2700}{12} = 225 \text{ m} \]

### Final Answer
The height of the tower is \( 225 \) m.

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