Determine the height of an antenna for a TV station that must be able to reach customers up to 80km away.
The correct answer and explanation is:
Correct Answer:
The required height of the antenna is approximately 502.5 meters.
Explanation:
The maximum distance dd (in kilometers) to the horizon from an antenna of height hh (in meters) can be approximated using the formula: d=3.57×hd = 3.57 \times \sqrt{h}
This formula assumes the Earth is a sphere and the observer is at sea level. To find the required antenna height to reach 80 kilometers, the equation must be rearranged to solve for hh: h=d3.57\sqrt{h} = \frac{d}{3.57} h=(803.57)2h = \left(\frac{80}{3.57}\right)^2 h=(22.41)2=502.5 metersh = (22.41)^2 = 502.5 \text{ meters}
So, the antenna needs to be approximately 502.5 meters tall to reach a distance of 80 kilometers.
This result is based on line-of-sight communication, which is typical for VHF and UHF television signals. These signals travel in straight lines and do not follow the curvature of the Earth. Because of this limitation, increasing the height of the transmitting antenna extends the visible range by allowing the signal to cover more ground before it curves out of sight due to Earth’s round surface.
This calculation assumes there are no significant obstacles such as mountains or buildings between the antenna and the receivers. In real-world settings, additional height may be necessary to overcome terrain variations or ensure signal strength over longer distances. This is why many TV stations place their antennas on towers, hills, or tall buildings to maximize their coverage area efficiently.