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Chapter 17: Problem 60

The area to be covered for TV telecast is doubled, then the height of transmitting antenna (TV tower) will have to be (A) Doubled (B) Halved (C) Quardupled (D) Kept unchanged

Short Answer

Expert verified
The height of the transmitting antenna (TV tower) will have to be quadrupled to cover double the area for TV telecast.

Step by step solution

01

Understand the relation between Line of Sight distance and height of antenna

We know that Line of Sight Distance is directly proportional to the square root of the height of the antenna. The formula is: \(D = k\sqrt{h}\) Where \(D\) is the Line of Sight Distance, \(h\) is the height of the antenna, and \(k\) is the proportionality constant. When the area is doubled, the new Line of Sight Distance will be twice the original distance. Let's denote the new height of the antenna as \(h_2\). Then, \(2D = k\sqrt{h_2}\). Now substitute the original Line of Sight Distance equation from the first equation into the second equation to find the relation between \(h\) and \(h_2\).
02

Finding the relation between the original height and the new height of the antenna

We have two equations: \(D = k\sqrt{h}\) \(2D = k\sqrt{h_2}\) Dividing the second equation by the first equation, we get \(\frac{2D}{D} = \frac{k\sqrt{h_2}}{k\sqrt{h}}\) Simplifying the above equation, we have: \(2 = \sqrt{\frac{h_2}{h}}\) Now, squaring both sides to eliminate the square root, we get: \(4 = \frac{h_2}{h}\) From the above equation, we can conclude the relation between the original height and the new height is: \(h_2 = 4h\)
03

Select the correct answer

Based on the relation we derived in step 2, we have \(h_2 = 4h\). This means the new height of the antenna will be four times the original height to cover double the area. Therefore, the correct answer is: (C) Quadrupled

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