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 how much the height of a TV tower needs to be increased to triple its coverage range, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Height of the tower (H1) = 100 m
- We need to find the new height (H2) that will triple the coverage range.

2. **Understand the Coverage Range Formula:**
- The coverage range (D) of a tower is given by the formula:
\[ D = \sqrt{2h \cdot r} \]
where \( h \) is the height of the tower and \( r \) is the radius of the Earth.

3. **Calculate the Initial Coverage Range (D1):**
- For the initial height (H1 = 100 m):
\[ D1 = \sqrt{2 \cdot 100 \cdot r} \]
- Here, \( r \) is the radius of the Earth, which is approximately \( 6400 \times 10^3 \) m.

4. **Set Up the New Coverage Range (D2):**
- If we want to triple the coverage range, then:
\[ D2 = 3 \cdot D1 \]

5. **Express D2 in Terms of the New Height (H2):**
- Using the coverage range formula for the new height (H2):
\[ D2 = \sqrt{2 \cdot H2 \cdot r} \]

6. **Equate the Two Coverage Range Expressions:**
- We have:
\[ 3 \cdot D1 = \sqrt{2 \cdot H2 \cdot r} \]
- Substitute \( D1 \) into the equation:
\[ 3 \cdot \sqrt{2 \cdot 100 \cdot r} = \sqrt{2 \cdot H2 \cdot r} \]

7. **Square Both Sides to Eliminate the Square Root:**
- Squaring both sides gives:
\[ 9 \cdot (2 \cdot 100 \cdot r) = 2 \cdot H2 \cdot r \]

8. **Simplify the Equation:**
- Cancel \( r \) from both sides (assuming \( r \neq 0 \)):
\[ 1800 = 2 \cdot H2 \]

9. **Solve for H2:**
- Dividing both sides by 2:
\[ H2 = 900 \text{ m} \]

10. **Calculate the Increase in Height:**
- The increase in height is given by:
\[ \text{Increase} = H2 - H1 = 900 - 100 = 800 \text{ m} \]

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