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 1: Understand the relationship between tower height and coverage range
The coverage range \( D \) of a TV tower is given by the formula:
\[ D = \sqrt{2 \cdot h_t \cdot r} \]
where \( h_t \) is the height of the tower and \( r \) is a constant related to the receiver's height (which we will assume to be zero for this problem).

### Step 2: Set the initial conditions
Given that the initial height of the tower \( h_t \) is 100 m, we can express the initial coverage range as:
\[ D = \sqrt{2 \cdot 100 \cdot r} = \sqrt{200r} \]

### Step 3: Determine the new coverage range
To triple the coverage range, we set the new coverage range \( D' \) to be:
\[ D' = 3D = 3 \cdot \sqrt{200r} \]

### Step 4: Relate the new coverage range to the new height
The new coverage range can also be expressed in terms of the new height \( h' \):
\[ D' = \sqrt{2 \cdot h' \cdot r} \]

### Step 5: Set up the equation
Equating the two expressions for \( D' \):
\[ 3 \cdot \sqrt{200r} = \sqrt{2 \cdot h' \cdot r} \]

### Step 6: Square both sides to eliminate the square root
Squaring both sides gives:
\[ (3 \cdot \sqrt{200r})^2 = ( \sqrt{2 \cdot h' \cdot r})^2 \]
\[ 9 \cdot 200r = 2 \cdot h' \cdot r \]

### Step 7: Simplify the equation
Dividing both sides by \( r \) (assuming \( r \neq 0 \)):
\[ 1800 = 2 \cdot h' \]

### Step 8: Solve for the new height \( h' \)
Now, solving for \( h' \):
\[ h' = \frac{1800}{2} = 900 \text{ m} \]

### Step 9: Calculate the increase in height
The increase in height \( \Delta h \) is given by:
\[ \Delta h = h' - h_t = 900 \text{ m} - 100 \text{ m} = 800 \text{ m} \]

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

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