Question

A TV transmitting antenna is 81 m tall. It has a half-power beam width of 10 degrees. If the receiving antenna is at the ground level, the service area covered by the transmitter is determined by

• A. beam width + height + radius
• B. height + radius
• C. beam width + radius
• D. height + beam width
• E. height

Hint:

Line-of-sight depends on Earth curvature.

Answer 1

The Correct Option is B

Concept: Distance to horizon
For an observer at height $h$ above the Earth's surface, the distance to the horizon is: \[ d = \sqrt{2Rh} \] where $R$ is the radius of the Earth.

Step 1: Geometrical idea
The line of sight to the horizon is tangential to the Earth. This forms a right-angled triangle with: - one side = $R$ (Earth’s radius) - other side = $R + h$

Step 2: Apply Pythagoras
\[ (R + h)^2 = R^2 + d^2 \]

Step 3: Simplify
\[ R^2 + 2Rh + h^2 = R^2 + d^2 \] \[ d^2 = 2Rh + h^2 \]

Step 4: Approximation
Since $h \ll R$, we neglect $h^2$: \[ d \approx \sqrt{2Rh} \] Final Answer:
\[ \boxed{d = \sqrt{2Rh}} \] Conclusion:
Distance to horizon depends on both Earth's radius $R$ and observer height $h$.