A spherical baloon of radius 50 cm, subtends an angle of `60^(@)` at a man's eye when the evevatin of its centre is `45^(@).` Then the height of the centre of the balloon is

Options

A. `50 sqrt 3`
B. `50 cm`
C. `40sqrt2 cm `
D. `50 sqrt2 cm`

Answer

To find the height of the center of the balloon, we can break down the problem into a few steps, using trigonometric relationships. Let's denote the following:

- Radius of the balloon, \( r = 50 \) cm
- Angle subtended at the man's eye, \( \theta = 60^\circ \)
- Angle of elevation to the center of the balloon, \( \alpha = 45^\circ \)

### Step 1: Understanding the Geometry
The balloon subtends an angle of \( 60^\circ \) at the man's eye. This means that from the man's eye level, the balloon appears to span an angle of \( 60^\circ \). Since the balloon is spherical, the angle can be divided into two equal angles of \( 30^\circ \) each from the center of the balloon to the edges.

### Step 2: Finding the Distance to the Balloon
Let’s denote the distance from the man’s eye to the center of the balloon as \( d \). In the right triangle formed by the radius of the balloon and the line of sight to the edge of the balloon, we can use the sine function:

\[ \sin(30^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{r}{d} \]

Substituting the known values:

\[ \sin(30^\circ) = \frac{50}{d} \]

Since \( \sin(30^\circ) = \frac{1}{2} \), we can set up the equation:

\[ \frac{1}{2} = \frac{50}{d} \]

### Step 3: Solving for \( d \)
Cross-multiplying gives:

\[ d = 100 \text{ cm} \]

### Step 4: Finding the Height to the Center of the Balloon
Now, we need to find the height \( H \) of the center of the balloon above the man's eye level. We can use the angle of elevation \( \alpha = 45^\circ \) in another right triangle formed between the man's eye, the center of the balloon, and the ground.

Using the sine function again:

\[ \sin(45^\circ) = \frac{H}{d} \]

Since \( \sin(45^\circ) = \frac{1}{\sqrt{2}} \):

\[ \frac{1}{\sqrt{2}} = \frac{H}{100} \]

### Step 5: Solving for \( H \)
Cross-multiplying gives:

\[ H = \frac{100}{\sqrt{2}} = 50\sqrt{2} \text{ cm} \]

### Step 6: Total Height of the Center of the Balloon
The total height of the center of the balloon above the ground is the height \( H \) plus the radius of the balloon:

\[ \text{Total Height} = H + r = 50\sqrt{2} + 50 \]

### Final Answer
Thus, the height of the center of the balloon above the ground is:

\[ \text{Total Height} = 50\sqrt{2} + 50 \text{ cm} \]