A anetenna is of height 500 m. What will be its range (Radio of the earth is 6400 km) ?

Options

A. 800 km
B. 100 km
C. 50 km
D. 80 km

Explanation

To find the range of an antenna based on its height, we can use the formula for the range \( r \) of a radio wave, which is given by:

\[ r = \sqrt{2 \cdot R_e \cdot h} \]

where:
- \( R_e \) is the radius of the Earth,
- \( h \) is the height of the antenna.

### Step 1: Convert the radius of the Earth to meters
The radius of the Earth is given as 6400 km. We need to convert this to meters:

\[ R_e = 6400 \text{ km} = 6400 \times 1000 \text{ m} = 6.4 \times 10^6 \text{ m} \]

### Step 2: Identify the height of the antenna
The height of the antenna is given as:

\[ h = 500 \text{ m} \]

### Step 3: Substitute values into the range formula
Now we can substitute the values of \( R_e \) and \( h \) into the range formula:

\[ r = \sqrt{2 \cdot (6.4 \times 10^6) \cdot 500} \]

### Step 4: Calculate the value inside the square root
First, calculate the product:

\[ 2 \cdot (6.4 \times 10^6) \cdot 500 = 2 \cdot 6.4 \cdot 500 \times 10^6 \]

Calculating \( 2 \cdot 6.4 \cdot 500 \):

\[ 2 \cdot 6.4 = 12.8 \]
\[ 12.8 \cdot 500 = 6400 \]

So, we have:

\[ 2 \cdot (6.4 \times 10^6) \cdot 500 = 6400 \times 10^6 \]

### Step 5: Take the square root
Now, we take the square root:

\[ r = \sqrt{6400 \times 10^6} = \sqrt{6400} \times \sqrt{10^6} \]

Calculating \( \sqrt{6400} \):

\[ \sqrt{6400} = 80 \]

And \( \sqrt{10^6} = 10^3 = 1000 \).

Thus, we have:

\[ r = 80 \times 1000 = 80000 \text{ m} \]

### Step 6: Convert the range to kilometers
Finally, convert the range from meters to kilometers:

\[ r = 80000 \text{ m} = 80 \text{ km} \]

### Conclusion
The range of the antenna is:

\[ \text{Range} = 80 \text{ km} \]