Problem 81
Question
Radio signal intensity Radio stations often have more than one broadcasting tower because federal guidelines do not usually permit a radio station to broadcast its signal in all directions with equal power. since radio waves can travel over long distances, it is important to control their directional patterns so that radio stations do not interfere with one another. Suppose that a radio station has two broadcasting towers located along a north-south line, as shown in the figure. If the radio station is broadcasting at a wavelength \(\lambda\) and the dis. tance between the two radio towers is equal to \(\frac{1}{2} \lambda\), then the intensity \(I\) of the signal in the direction \(\theta\) is given by $$I=\frac{1}{2} I_{0}[1+\cos (\pi \sin \theta)]$$ where \(I_{0}\) is the maximum intensity. Approximate \(I\) in terms of \(I_{0}\) for each \(\theta .\) (a) \(\theta=0\) (b) \(\quad \theta=\pi / 3\) (c) \(\theta=\pi / 7\)
Step-by-Step Solution
Verified Answer
(a) \( I = I_{0} \), (b) \( I \approx 0.773 I_{0} \), (c) \( I \approx 0.965 I_{0} \).
1Step 1: Calculate Intensity for θ = 0
Substitute \( \theta = 0 \) into the intensity equation: \[I = \frac{1}{2} I_{0} \left[ 1 + \cos (\pi \sin 0) \right] = \frac{1}{2} I_{0} [1 + \cos (0)]\]\[I = \frac{1}{2} I_{0} [1 + 1] = \frac{1}{2} I_{0} \times 2 = I_{0}\]Thus, for \( \theta = 0 \), the intensity \( I = I_{0} \).
2Step 2: Calculate Intensity for θ = π/3
Substitute \( \theta = \frac{\pi}{3} \) into the intensity equation:\[I = \frac{1}{2} I_{0} \left[ 1 + \cos \left(\pi \sin \frac{\pi}{3} \right) \right]\]Compute \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \): \[I = \frac{1}{2} I_{0} \left[ 1 + \cos \left( \frac{\pi \sqrt{3}}{2} \right) \right]\]The value for \( \cos \left( \frac{\pi \sqrt{3}}{2} \right) \) is needed, requiring a calculator, but conceptually this is what you substitute in to solve numerically.
3Step 3: Calculate Intensity for θ = π/7
Substitute \( \theta = \frac{\pi}{7} \) into the intensity equation:\[I = \frac{1}{2} I_{0} \left[ 1 + \cos \left( \pi \sin \frac{\pi}{7} \right) \right]\]Compute \( \sin \frac{\pi}{7} \), and substitute in back into the formula to find \( \cos \left( \pi \sin \frac{\pi}{7} \right) \). More precise values require calculator evaluations or approximations, but the formula reduces to a computable numeric form for a calculator.
Key Concepts
Radio Signal IntensityBroadcasting TowersTrigonometric FunctionsSignal Directional Patterns
Radio Signal Intensity
Radio signal intensity measures the power of the radio signal received at a particular point. It is influenced by several factors, including the power of the broadcasting station and the distance from the broadcasting towers. Intensity is crucial because it determines whether the signal remains clear and strong as it reaches listeners.
Signals that are too weak can be overshadowed by interference, while overly intense signals can cause network congestion. Radio signal intensity can be precisely controlled using mathematical equations such as the Trigonometric Intensity Equation. This equation considers not only the distance but also the angle, or direction, at which the signal is transmitted, thus ensuring optimal signal strength and clarity at various locations.
Signals that are too weak can be overshadowed by interference, while overly intense signals can cause network congestion. Radio signal intensity can be precisely controlled using mathematical equations such as the Trigonometric Intensity Equation. This equation considers not only the distance but also the angle, or direction, at which the signal is transmitted, thus ensuring optimal signal strength and clarity at various locations.
Broadcasting Towers
Broadcasting towers are essential for transmitting signals over long distances. They serve as the launching point for radio waves that are sent to receivers in homes, cars, and other devices. These towers work together in networks to cover broader areas and enhance signal bandwidth.
Federal guidelines often restrict the broadcast range to prevent radio waves from interfering with each other. As a result, stations may use multiple towers strategically placed. By managing the placement and power output of broadcasting towers, stations can control the signal patterns effectively, thereby reducing overlap and maximizing coverage area.
Federal guidelines often restrict the broadcast range to prevent radio waves from interfering with each other. As a result, stations may use multiple towers strategically placed. By managing the placement and power output of broadcasting towers, stations can control the signal patterns effectively, thereby reducing overlap and maximizing coverage area.
Trigonometric Functions
Trigonometric functions such as sine and cosine are fundamental in calculating the directional intensity of radio signals. They help to model the wave patterns that determine how signals propagate from broadcasting towers.
In the Trigonometric Intensity Equation, these functions help ascertain how much of the signal intensity should be directed along a particular angle.
For example, a scenario where the angle is zero (\(\theta = 0)\) would use these functions to show that the maximum intensity is in that direction. Trigonometry thus ensures that the resultant signal broadcasted is optimally directed and distributed.
In the Trigonometric Intensity Equation, these functions help ascertain how much of the signal intensity should be directed along a particular angle.
For example, a scenario where the angle is zero (\(\theta = 0)\) would use these functions to show that the maximum intensity is in that direction. Trigonometry thus ensures that the resultant signal broadcasted is optimally directed and distributed.
Signal Directional Patterns
Signal directional patterns define how radio waves are distributed from the broadcasting towers. They help in managing how signals are sent out to cover intended coverage areas effectively.
The patterns dictate the primary path of the radio waves, indicating which directions receive more signal power and which receive less. This precise control helps avoid signal interference with other stations broadcasting nearby and ensures consistent signal quality within the designated transmission area.
The patterns dictate the primary path of the radio waves, indicating which directions receive more signal power and which receive less. This precise control helps avoid signal interference with other stations broadcasting nearby and ensures consistent signal quality within the designated transmission area.
- Signal patterns are especially useful in congested areas where multiple stations operate.
- Direction patterns help enhance signal clarity by focusing power to where it's needed more.
Other exercises in this chapter
Problem 80
Use a graph to solve the inequality on the interval \([-\pi, \pi]\) $$\frac{1}{2} \cos 2 x+2 \cos (x-2)
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As \(x \rightarrow 0^{+}, f(x) \rightarrow L\) for some real number \(L\) Use a graph to predict \(L\) $$f(x)=\frac{x+\tan x}{\sin x}$$
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