Question

When the receiving antenna is on the ground, the range of a transmitting antenna of height 980 m is (Radius of the earth = 6400 km)

• A. 56 km
• B. 112 km
• C. 72.4 km
• D. 224 km

Hint:

When using the formula \(d = \sqrt{2Rh}\), if you keep R in km and h in meters, a useful approximation is \(d(\text{km}) \approx \sqrt{2 \times 6400 \times h/1000} = \sqrt{12.8h}\). For this problem: \(d \approx \sqrt{12.8 \times 980} = \sqrt{12544} = 112\) km. This avoids dealing with large powers of 10.

Answer 1

The Correct Option is B

Step 1: Formula for Range (Horizon Distance): The distance to the horizon (range \( d \)) from an antenna of height \( h \) is given by: \[ d = \sqrt{2Rh} \] where \( R \) is the radius of the Earth. (Assuming receiving antenna height is negligible/on ground).
Step 2: Substitute Values: \( h = 980 \, \text{m} \) \( R = 6400 \, \text{km} = 6400 \times 10^3 \, \text{m} = 6.4 \times 10^6 \, \text{m} \) \[ d = \sqrt{2 \times (6.4 \times 10^6) \times 980} \] \[ d = \sqrt{12.8 \times 98 \times 10^7} \] Let's simplify differently: \[ d = \sqrt{2 \times 6400 \times 1000 \times \frac{980}{1000} \times 1000} \] Wait, simpler calculation: \[ d = \sqrt{2 \times 64 \times 10^5 \times 9.8 \times 10^2} \] \[ d = \sqrt{2 \times 6.4 \times 10^6 \times 9.8 \times 10^2 \times 10^{-2}} \] No. Direct multiply: \( 2 \times 6400000 \times 980 = 12800000 \times 980 \) \( 12.8 \times 10^6 \times 980 = 12544 \times 10^6 \) \( d = \sqrt{12544 \times 10^6} = \sqrt{12544} \times 10^3 \) We know \( \sqrt{12544} = 112 \) (since \( 112^2 = (100+12)^2 = 10000 + 144 + 2400 = 12544 \)). \[ d = 112 \times 10^3 \, \text{m} = 112 \, \text{km} \]
Step 3: Final Answer: The range is 112 km.