A TV tower has a height of 100m. By how much the height of tower be increased to triple it coverage range.

Answer

To solve the problem of determining how much the height of a TV tower must be increased to triple its coverage range, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship Between Height and Range**:
The coverage range (d) of a TV tower is related to its height (h) by the formula:
\[ d^2 = 2rh \]
where \( r \) is the radius of the Earth.

2. **Initial Conditions**:
Given that the initial height of the tower \( h_1 = 100 \) m, we can express the initial range as \( d_1 \).

3. **Tripling the Coverage Range**:
We want to triple the coverage range, so the new range will be:
\[ d_2 = 3d_1 \]

4. **Using the Relationship**:
Substitute \( d_2 \) into the relationship:
\[ d_2^2 = 2rh_2 \]
where \( h_2 \) is the new height of the tower.

5. **Setting Up the Equation**:
From the original and new conditions, we can write:
\[ (3d_1)^2 = 2rh_2 \]
Expanding this gives:
\[ 9d_1^2 = 2rh_2 \]

6. **Relating the Two Equations**:
We know from the original height that:
\[ d_1^2 = 2rh_1 \]
Substitute \( d_1^2 \) into the equation for \( d_2^2 \):
\[ 9(2rh_1) = 2rh_2 \]

7. **Simplifying the Equation**:
Cancel \( 2r \) from both sides (assuming \( r \neq 0 \)):
\[ 9h_1 = h_2 \]

8. **Calculating the New Height**:
Substitute \( h_1 = 100 \) m:
\[ h_2 = 9 \times 100 = 900 \text{ m} \]

9. **Finding the Increase in Height**:
The increase in height required is:
\[ \text{Increase} = h_2 - h_1 = 900 - 100 = 800 \text{ m} \]

### Final Answer:
The height of the tower must be increased by **800 meters** to triple its coverage range.

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