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. 6
B. 7
C. 8
D. 9

Explanation

To solve the problem step by step, we will use the information provided in the question and apply trigonometric principles.

### Step 1: Understand the Problem
We have a tower PT of height \(2^x\) meters. Points A and B are at distances \(AP = 2^{(x+1)}\) meters and \(BP = 192\) meters respectively. The angle of elevation from B is double that from A.

### Step 2: Set Up the Trigonometric Relationships
From point A:
- The angle of elevation is \(\theta\).
- Using the tangent function, we have:
\[ \tan(\theta) = \frac{PT}{AP} = \frac{2^x}{2^{(x+1)}} = \frac{2^x}{2 \cdot 2^x} = \frac{1}{2} \]

From point B:
- The angle of elevation is \(2\theta\).
- Using the tangent function, we have:
\[ \tan(2\theta) = \frac{PT}{BP} = \frac{2^x}{192} \]

### Step 3: Use the Double Angle Formula for Tangent
The double angle formula for tangent is:
\[ \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} \]
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 4: Set Up the Equation
Now we equate the two expressions for \(\tan(2\theta)\):
\[ \frac{2^x}{192} = \frac{4}{3} \]

### Step 5: Cross Multiply to Solve for \(2^x\)
Cross multiplying gives:
\[ 3 \cdot 2^x = 4 \cdot 192 \]
Calculating \(4 \cdot 192\):
\[ 4 \cdot 192 = 768 \]
So we have:
\[ 3 \cdot 2^x = 768 \]

### Step 6: Divide by 3
Dividing both sides by 3:
\[ 2^x = \frac{768}{3} = 256 \]

### Step 7: Solve for \(x\)
Since \(256 = 2^8\), we can equate:
\[ 2^x = 2^8 \implies x = 8 \]

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

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