PT. a tower of height `2^(x)` metre, P being the foot. T being the top of the tower. A, B are points on the same line with P. If `AP = 2^(x+1)m. BP = 192 m` and if the angle of elevation of the tower as seen from b is double the angle of the elevation of the tower as seen from A, then what is the value of x?

Options

A. A. 6
B. B. 7
C. C. 8
D. D. 9

Answer

To solve the problem step by step, we will follow the given information and apply trigonometric principles to find the value of \( x \).

### Step 1: Understand the Problem
We have a tower \( PT \) with height \( 2^x \) meters, where \( P \) is the foot of the tower. The distance from point \( A \) to point \( P \) is \( AP = 2^{x+1} \) meters, and the distance from point \( B \) to point \( P \) is \( BP = 192 \) meters. The angle of elevation from point \( B \) is double that from point \( A \).

### Step 2: Set Up the Angles
Let the angle of elevation from point \( A \) be \( \theta \). Therefore, the angle of elevation from point \( B \) will be \( 2\theta \).

### Step 3: Use Triangle \( PTA \)
In triangle \( PTA \):
- The height \( PT = 2^x \) (the height of the tower).
- The base \( AP = 2^{x+1} \).

Using the tangent function:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{2^x}{2^{x+1}} = \frac{2^x}{2^x \cdot 2} = \frac{1}{2} \]
This gives us our first equation:
\[ \tan(\theta) = \frac{1}{2} \]

### Step 4: Use Triangle \( PTB \)
In triangle \( PTB \):
- The height \( PT = 2^x \).
- The base \( BP = 192 \).

Using the tangent function:
\[ \tan(2\theta) = \frac{2^x}{192} \]

### Step 5: Use the Double Angle Formula
We know from trigonometry that:
\[ \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} \]
Substituting \( \tan(\theta) = \frac{1}{2} \):
\[ \tan(2\theta) = \frac{2 \cdot \frac{1}{2}}{1 - \left(\frac{1}{2}\right)^2} = \frac{1}{1 - \frac{1}{4}} = \frac{1}{\frac{3}{4}} = \frac{4}{3} \]

### Step 6: Set Up the Equation
Now we can set up the equation using the value of \( \tan(2\theta) \):
\[ \frac{4}{3} = \frac{2^x}{192} \]

### Step 7: Solve for \( 2^x \)
Cross-multiplying gives:
\[ 4 \cdot 192 = 3 \cdot 2^x \]
Calculating \( 4 \cdot 192 \):
\[ 768 = 3 \cdot 2^x \]
Dividing both sides by 3:
\[ 2^x = \frac{768}{3} = 256 \]

### Step 8: Solve for \( x \)
Recognizing that \( 256 = 2^8 \):
\[ 2^x = 2^8 \implies x = 8 \]

### Final Answer
The value of \( x \) is \( 8 \).

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