A TV tower has a height of `150 m`. The area of the region covered by the TV broadcast is (Radius of earth `= 6.4 xx 10^(6) m` )

Options

A. `9.6pi xx 10^(8)m^(2)`
B. `19.2pi xx 10^(7)m^(2)`
C. `19.2pi xx 10^(10)m^(2)`
D. `19.2pi xx 10^(2)km^(2)`

Answer

To solve the problem of finding the area covered by the TV broadcast from a tower of height 150 m, we can follow these steps:

### Step 1: Understand the Geometry
We need to find the distance (d) from the top of the tower to the horizon. This can be visualized as a right triangle where:
- One leg is the height of the tower (h = 150 m).
- The other leg is the radius of the Earth (r = 6.4 × 10^6 m).
- The hypotenuse is the distance from the top of the tower to the horizon.

### Step 2: Use the Pythagorean Theorem
According to the Pythagorean theorem, we can express the relationship between these distances as:
\[ d = \sqrt{2hr} \]
This formula is derived under the assumption that the height of the tower is much smaller than the radius of the Earth.

### Step 3: Substitute Known Values
Now, we can substitute the known values into the formula:
- Height of the tower, \( h = 150 \, \text{m} \)
- Radius of the Earth, \( r = 6.4 \times 10^6 \, \text{m} \)

So, we calculate:
\[ d = \sqrt{2 \times 150 \, \text{m} \times 6.4 \times 10^6 \, \text{m}} \]

### Step 4: Calculate d
Calculating the value inside the square root:
\[ d = \sqrt{2 \times 150 \times 6.4 \times 10^6} \]
\[ = \sqrt{1920 \times 10^6} \]
\[ = 1385.615 \, \text{m} \] (approximately)

### Step 5: Calculate the Area
The area (A) covered by the broadcast is given by the formula:
\[ A = \pi d^2 \]
Substituting the value of d:
\[ A = \pi (1385.615)^2 \]

### Step 6: Calculate A
Calculating the area:
\[ A = \pi \times 1920 \times 10^6 \]
\[ A \approx 1920 \pi \, \text{m}^2 \]

### Step 7: Convert to Kilometers
To convert the area from square meters to square kilometers:
\[ A \approx 1920 \pi \times 10^{-6} \, \text{km}^2 \]
\[ = 19.20 \pi \times 10^2 \, \text{km}^2 \]

### Final Answer
Thus, the area covered by the TV broadcast is:
\[ A \approx 19.20 \pi \times 10^2 \, \text{km}^2 \]

---