At a point on a horizontal line through the base of a monument the angle of elevation of the top of the monument of the top of the monument is found to be such that its tangent is `(1)/(5)` . On walking 138 metres towards the monument the secant of the angle of elevation is found to be `(sqrt(193))/(12)` .The height of the monument (in metre ) is

Options

A. 35
B. 49
C. 42
D. 56

Answer

To solve the problem step by step, we will use trigonometric relationships based on the information provided.

### Step 1: Define the Variables
Let:
- \( h \) = height of the monument
- \( x \) = distance from the initial point to the base of the monument

### Step 2: Use the Tangent Function
From the problem, we know that the tangent of the angle of elevation at the initial point is \( \frac{1}{5} \). This gives us the equation:
\[ \tan(\alpha) = \frac{h}{x} \]
Since \( \tan(\alpha) = \frac{1}{5} \), we can write:
\[ \frac{h}{x} = \frac{1}{5} \]
This implies:
\[ h = \frac{1}{5}x \quad \text{(Equation 1)} \]

### Step 3: Move Towards the Monument
After walking 138 meters towards the monument, the new distance to the base of the monument becomes \( x - 138 \).

### Step 4: Use the Secant Function
The secant of the angle of elevation at the new position is given as \( \sec(\beta) = \frac{\sqrt{193}}{12} \). We can find \( \tan(\beta) \) using the identity:
\[ \sec^2(\beta) = 1 + \tan^2(\beta) \]
Calculating \( \sec^2(\beta) \):
\[ \sec^2(\beta) = \left(\frac{\sqrt{193}}{12}\right)^2 = \frac{193}{144} \]
Thus,
\[ 1 + \tan^2(\beta) = \frac{193}{144} \]
This leads to:
\[ \tan^2(\beta) = \frac{193}{144} - 1 = \frac{193 - 144}{144} = \frac{49}{144} \]
Taking the square root gives:
\[ \tan(\beta) = \frac{7}{12} \quad \text{(Equation 2)} \]

### Step 5: Set Up the Second Tangent Equation
From the second position, we have:
\[ \tan(\beta) = \frac{h}{x - 138} \]
Substituting the value of \( \tan(\beta) \):
\[ \frac{h}{x - 138} = \frac{7}{12} \]
This implies:
\[ h = \frac{7}{12}(x - 138) \quad \text{(Equation 3)} \]

### Step 6: Solve the Equations
Now we have two equations:
1. \( h = \frac{1}{5}x \) (from Equation 1)
2. \( h = \frac{7}{12}(x - 138) \) (from Equation 3)

Setting these equal to each other:
\[ \frac{1}{5}x = \frac{7}{12}(x - 138) \]

### Step 7: Clear the Fractions
To eliminate the fractions, multiply through by the least common multiple (60):
\[ 12x = 35(x - 138) \]
Expanding the right side:
\[ 12x = 35x - 4830 \]
Rearranging gives:
\[ 35x - 12x = 4830 \]
\[ 23x = 4830 \]
\[ x = \frac{4830}{23} = 210 \]

### Step 8: Find the Height \( h \)
Now substitute \( x \) back into Equation 1 to find \( h \):
\[ h = \frac{1}{5}(210) = 42 \text{ meters} \]

### Final Answer
The height of the monument is \( \boxed{42} \) meters.