At a distance 12 metres from the foot A of a tower AB of height 5 metres, a flagstaff BC on top of AB and the tower subtend the same angle. Then, the height of flagstaff is

Options

A. `1440/199` metres
B. `475/119` metres
C. `845/119` metres
D. none of these

Answer

To solve the problem step by step, let's break it down clearly:

### Given:
- Height of the tower \( AB = 5 \) meters
- Distance from the foot of the tower \( A \) to point \( D = 12 \) meters
- Let the height of the flagstaff \( BC = x \) meters

### Step 1: Understand the Angles
The angles subtended by the tower \( AB \) and the flagstaff \( BC \) at point \( D \) are the same. Let this angle be \( \theta \).

### Step 2: Set Up the Triangle for the Tower
In triangle \( ABD \):
- The height of the tower \( AB = 5 \) meters (perpendicular)
- The distance from \( A \) to \( D = 12 \) meters (base)

Using the tangent function:
\[ \tan(\theta) = \frac{AB}{AD} = \frac{5}{12} \]

### Step 3: Calculate \( \theta \)
From the above equation, we can express \( \theta \) as:
\[ \theta = \tan^{-1}\left(\frac{5}{12}\right) \]

### Step 4: Set Up the Triangle for the Flagstaff
Now consider triangle \( CAD \):
- The height of the flagstaff plus the tower is \( AC = AB + BC = 5 + x \)
- The distance from \( A \) to \( D = 12 \) meters (base)

Using the tangent function again:
\[ \tan(\theta) = \frac{AC}{AD} = \frac{5 + x}{12} \]

### Step 5: Use the Identity for Tangent of Double Angle
Since both angles are equal, we can use the double angle formula:
\[ \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} \]
Substituting \( \tan(\theta) = \frac{5}{12} \):
\[ \tan(2\theta) = \frac{2 \cdot \frac{5}{12}}{1 - \left(\frac{5}{12}\right)^2} \]
Calculating \( \tan^2(\theta) \):
\[ \tan^2(\theta) = \left(\frac{5}{12}\right)^2 = \frac{25}{144} \]
Thus,
\[ 1 - \tan^2(\theta) = 1 - \frac{25}{144} = \frac{119}{144} \]
Now substituting back:
\[ \tan(2\theta) = \frac{\frac{10}{12}}{\frac{119}{144}} = \frac{10 \cdot 144}{12 \cdot 119} = \frac{120}{119} \]

### Step 6: Set Up the Equation
From triangle \( CAD \):
\[ \tan(2\theta) = \frac{5 + x}{12} \]
Setting the two expressions for \( \tan(2\theta) \) equal gives:
\[ \frac{5 + x}{12} = \frac{120}{119} \]

### Step 7: Solve for \( x \)
Cross-multiplying to solve for \( x \):
\[ 119(5 + x) = 120 \cdot 12 \]
\[ 119(5 + x) = 1440 \]
\[ 5 + x = \frac{1440}{119} \]
\[ x = \frac{1440}{119} - 5 \]
Calculating \( 5 \) in terms of \( 119 \):
\[ 5 = \frac{595}{119} \]
Thus,
\[ x = \frac{1440 - 595}{119} = \frac{845}{119} \]

### Final Answer
The height of the flagstaff \( BC \) is:
\[ x = \frac{845}{119} \text{ meters} \]