The area to be covered for TV telecast is doubled. Then the height of the transmitting antenna will have to be

Options

A. doubled
B. halved
C. quadrupled
D. kept unchanged

Answer

To solve the problem of how the height of the transmitting antenna changes when the area to be covered for TV telecast is doubled, we can follow these steps:

### Step 1: Understand the relationship between area and height
The area \( A \) covered by the transmitting antenna can be expressed in terms of the range \( D \) of the signal:
\[ A = \pi D^2 \]
where \( D \) is the range of the antenna.

### Step 2: Relate the range to the height of the antenna
The range \( D \) can be approximated using the formula:
\[ D = \sqrt{2RH} \]
where \( R \) is the radius of the Earth and \( H \) is the height of the transmitting antenna.

### Step 3: Substitute the range into the area formula
Substituting \( D \) into the area formula gives:
\[ A = \pi (2RH) = 2\pi RH \]

### Step 4: Set up the equation for the new area
If the area is doubled, we denote the new area as \( 2A \):
\[ 2A = 2\pi RH_2 \]
where \( H_2 \) is the new height of the antenna.

### Step 5: Relate the old area to the new area
From the original area, we have:
\[ A = 2\pi RH_1 \]
where \( H_1 \) is the original height of the antenna.

### Step 6: Create a ratio of the areas
Setting up the ratio of the areas gives:
\[ \frac{A}{2A} = \frac{2\pi RH_1}{2\pi RH_2} \]
This simplifies to:
\[ \frac{1}{2} = \frac{H_1}{H_2} \]

### Step 7: Solve for the new height
Rearranging the equation gives:
\[ H_2 = 2H_1 \]

### Conclusion
Thus, when the area to be covered for TV telecast is doubled, the height of the transmitting antenna must also be doubled.

### Final Answer:
The height of the transmitting antenna will have to be doubled.

---