TV tower has a height of 80m. The height of the tower to be increased to, so that the coverage area is doubled is

Options

A. 80m
B. 160m
C. 240m
D. 320 m

Answer

To solve the problem of determining the new height of a TV tower so that its coverage area is doubled, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship Between Height and Coverage Area**:
The coverage area of a TV tower is related to its height. The range \( d \) of the coverage can be expressed as:
\[ d = \sqrt{2 \cdot R \cdot h} \]
where \( R \) is the radius of the Earth and \( h \) is the height of the tower.

2. **Initial Height of the Tower**:
The initial height of the tower \( h \) is given as:
\[ h = 80 \text{ m} \]

3. **Determine the New Coverage Area**:
If the coverage area is to be doubled, the new range \( d' \) will be:
\[ d' = 2d \]

4. **Express the New Range in Terms of Height**:
Using the formula for range, we can express the new range as:
\[ d' = \sqrt{2 \cdot R \cdot h'} \]
where \( h' \) is the new height of the tower.

5. **Set Up the Equation**:
Since \( d' = 2d \), we can square both sides to eliminate the square root:
\[ (2d)^2 = d'^2 \]
This gives us:
\[ 4d^2 = 2Rh' \]
We also know:
\[ d^2 = 2Rh \]
Therefore, substituting \( d^2 \) into the equation gives:
\[ 4(2Rh) = 2Rh' \]

6. **Simplify the Equation**:
Simplifying the equation:
\[ 8Rh = 2Rh' \]
Dividing both sides by \( 2R \) (assuming \( R \neq 0 \)):
\[ 4h = h' \]

7. **Calculate the New Height**:
Now, substituting the value of \( h \):
\[ h' = 4 \cdot 80 \text{ m} = 320 \text{ m} \]

8. **Determine the Increase in Height**:
The increase in height is given by:
\[ \text{Increase in height} = h' - h = 320 \text{ m} - 80 \text{ m} = 240 \text{ m} \]

### Final Answer:
The height of the tower needs to be increased by **240 meters**.

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