To double the covering range of a TV transmittion tower, its height should be multiplied by :

Options

A. `(1)/(sqrt(2))`
B. 2
C. 4
D. `sqrt(2)`

Explanation

To solve the problem of determining how much the height of a TV transmission tower should be multiplied to double its covering range, we can follow these steps:

### Step 1: Understand the relationship between height and range
The covering range \( d \) of a TV transmission tower is given by the formula:
\[ d = \sqrt{2 \cdot h_t \cdot R} \]
where:
- \( h_t \) is the height of the transmitting antenna,
- \( R \) is the radius of the Earth (approximately 6400 km).

### Step 2: Set up the equation for the original range
Let’s denote the original height of the tower as \( h_t \). The original covering range can be expressed as:
\[ d = \sqrt{2 \cdot h_t \cdot R} \]

### Step 3: Set up the equation for the new range
If we want to double the covering range, the new range \( d' \) will be:
\[ d' = 2d \]
Substituting the expression for \( d \):
\[ d' = 2 \sqrt{2 \cdot h_t \cdot R} \]

### Step 4: Express the new range in terms of height
Using the same formula for the new height \( h_t' \):
\[ d' = \sqrt{2 \cdot h_t' \cdot R} \]
Setting the two expressions for \( d' \) equal gives:
\[ 2 \sqrt{2 \cdot h_t \cdot R} = \sqrt{2 \cdot h_t' \cdot R} \]

### Step 5: Square both sides to eliminate the square root
Squaring both sides results in:
\[ (2 \sqrt{2 \cdot h_t \cdot R})^2 = ( \sqrt{2 \cdot h_t' \cdot R})^2 \]
This simplifies to:
\[ 4 \cdot 2 \cdot h_t \cdot R = 2 \cdot h_t' \cdot R \]
\[ 8 \cdot h_t \cdot R = 2 \cdot h_t' \cdot R \]

### Step 6: Cancel \( R \) from both sides
Assuming \( R \) is not zero, we can divide both sides by \( R \):
\[ 8 \cdot h_t = 2 \cdot h_t' \]

### Step 7: Solve for the new height \( h_t' \)
Now, divide both sides by 2:
\[ h_t' = 4 \cdot h_t \]

### Conclusion
To double the covering range of a TV transmission tower, its height should be multiplied by **4**.

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