A spherical gas balloon of radius 16 meter subtends an angle `60^(@)` at the eye of the observer A while the angle of elevation of its center from the eye of A is `75^(@)`. Then the height (in meter) of the top most point of the balloon from the level of the observer's eye is :

Options

A. `8(2+2sqrt(3)+sqrt(2))`
B. `8(sqrt(6)+sqrt(2)+2)`
C. `8(sqrt(2)+2+sqrt(3))`
D. `8(sqrt(6)-sqrt(2)+2)`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Geometry
We have a spherical balloon with a radius of 16 meters. The balloon subtends an angle of \(60^\circ\) at the observer's eye, and the angle of elevation to the center of the balloon is \(75^\circ\).

### Step 2: Draw the Diagram
Draw a diagram to visualize the situation:
- Let \(O\) be the center of the balloon.
- Let \(A\) be the observer's eye level.
- Let \(B\) be the topmost point of the balloon.
- Let \(P\) be the point directly below \(B\) on the ground.

### Step 3: Determine Angles
From the information given:
- The angle subtended at the observer's eye by the balloon is \(60^\circ\), meaning the angle \(QPR\) is \(60^\circ\).
- The angle of elevation to the center \(O\) from point \(A\) is \(75^\circ\).

### Step 4: Find the Height of the Center of the Balloon
Let \(h\) be the height of the center \(O\) from the observer's eye level \(A\). We can use the triangle formed by the observer and the center of the balloon.

Using the tangent of the angle of elevation:
\[ \tan(75^\circ) = \frac{h}{d} \]
where \(d\) is the horizontal distance from the observer to the point directly below the center of the balloon.

### Step 5: Find the Horizontal Distance
Using the angle subtended by the balloon:
- The angle at the observer's eye is \(60^\circ\), which means the angle from the center of the balloon to the edges is \(30^\circ\) on either side (since \(60^\circ/2 = 30^\circ\)).
- Therefore, we can relate the radius of the balloon to the distance \(d\):
\[ \tan(30^\circ) = \frac{16}{d} \]
From this, we can find \(d\):
\[ d = \frac{16}{\tan(30^\circ)} = 16 \cdot \sqrt{3} \]

### Step 6: Calculate the Height \(h\)
Now, substituting \(d\) back into the equation for \(h\):
\[ h = d \cdot \tan(75^\circ) = (16 \cdot \sqrt{3}) \cdot \tan(75^\circ) \]

### Step 7: Find the Height of the Topmost Point \(B\)
The height of the topmost point \(B\) from the observer's eye level \(A\) is:
\[ AB = OA + OB \]
Where \(OA\) is the radius of the balloon (16 meters) and \(OB\) is the height \(h\) we calculated.

### Step 8: Final Calculation
Now, we can calculate the total height:
\[ AB = 16 + h \]
Substituting the value of \(h\) gives us the final answer.