A radar has a power of `1kW` and is operating at a frequency of `10 GHz`. It is located on a mountain top of height `500m`. The maximum distance upto which it can detect object located on the surface of the earth (Radius of earth `6.4xx10^(6)m`) is

Options

A. 80 km
B. 16 km
C. 40 km
D. 64 km

Answer

To solve the problem of determining the maximum distance up to which a radar can detect an object located on the surface of the Earth, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values**:
- Power of the radar: \( P = 1 \, \text{kW} \) (not directly needed for this calculation)
- Frequency of the radar: \( f = 10 \, \text{GHz} \) (not directly needed for this calculation)
- Height of the mountain: \( h = 500 \, \text{m} = 0.5 \, \text{km} \)
- Radius of the Earth: \( R = 6.4 \times 10^6 \, \text{m} = 6400 \, \text{km} \)

2. **Set Up the Geometry**:
- The radar is located at a height \( h \) above the Earth's surface. The distance \( d \) we want to find is the horizontal distance from the base of the mountain to the point where the radar can detect an object on the surface of the Earth.
- The situation can be visualized as a right triangle where:
- One leg is the radius of the Earth \( R \).
- The other leg is the height \( h \).
- The hypotenuse is \( R + h \).

3. **Apply Pythagorean Theorem**:
- According to the Pythagorean theorem:
\[ (R + h)^2 = R^2 + d^2 \]
- Expanding the left side:
\[ R^2 + 2Rh + h^2 = R^2 + d^2 \]

4. **Simplify the Equation**:
- Cancel \( R^2 \) from both sides:
\[ 2Rh + h^2 = d^2 \]
- Since \( h \) is much smaller than \( R \), we can ignore \( h^2 \):
\[ d^2 \approx 2Rh \]

5. **Calculate the Distance \( d \)**:
- Substitute the values of \( R \) and \( h \):
\[ d = \sqrt{2Rh} \]
- Convert \( R \) to kilometers:
\[ R = 6400 \, \text{km} = 6.4 \times 10^6 \, \text{m} \]
- Substitute \( R = 6400 \, \text{km} \) and \( h = 0.5 \, \text{km} \):
\[ d = \sqrt{2 \times 6400 \times 0.5} \]
\[ d = \sqrt{6400} = 80 \, \text{km} \]

6. **Final Result**:
- The maximum distance up to which the radar can detect an object located on the surface of the Earth is \( d = 80 \, \text{km} \).