A TV tower has a height of 75m. What is the maximum distance upto which this TV transmission can be received ? (Radius of the earth = `6.4 xx 10^(6)`m)

Options

A. 30.98 km
B. 38.98 km
C. 40.98 km
D. 50.98 km

Answer

To find the maximum distance up to which the TV transmission can be received from a tower of height 75 meters, we can use the formula for the maximum distance of transmission:

\[ d = \sqrt{2 \cdot R \cdot h} \]

where:
- \( d \) is the maximum distance,
- \( R \) is the radius of the Earth,
- \( h \) is the height of the transmitting antenna.

### Step 1: Identify the given values
- Height of the TV tower, \( h = 75 \, \text{m} \)
- Radius of the Earth, \( R = 6.4 \times 10^6 \, \text{m} \)

### Step 2: Substitute the values into the formula
Using the formula, we substitute the values of \( R \) and \( h \):

\[ d = \sqrt{2 \cdot (6.4 \times 10^6) \cdot 75} \]

### Step 3: Calculate the product inside the square root
First, calculate \( 2 \cdot 6.4 \times 10^6 \):

\[ 2 \cdot 6.4 = 12.8 \]

Now multiply by 75:

\[ 12.8 \cdot 75 = 960 \]

Now we have:

\[ d = \sqrt{960 \times 10^6} \]

### Step 4: Simplify the expression
This can be rewritten as:

\[ d = \sqrt{960} \cdot \sqrt{10^6} \]

Since \( \sqrt{10^6} = 10^3 = 1000 \):

\[ d = \sqrt{960} \cdot 1000 \]

### Step 5: Calculate \( \sqrt{960} \)
Calculating \( \sqrt{960} \):

\[ \sqrt{960} \approx 31.0 \]

### Step 6: Final calculation for distance
Now, substituting back:

\[ d \approx 31.0 \cdot 1000 = 31000 \, \text{m} \]

### Step 7: Convert meters to kilometers
To convert meters to kilometers:

\[ d \approx 31.0 \, \text{km} \]

### Conclusion
The maximum distance up to which the TV transmission can be received is approximately **31.0 kilometers**.

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