Assertion: When the height of a TV transmission tower is increased by three times. The range covered is doubled.
Reason: The range covered is proportional to the height of the TV transmission tower.

Options

A. A and R are true and R is the correct explanation
B. Both A and R aer true but reason is not the correct explanation
C. a is true R is false
D. Both A and R are true

Answer

To analyze the given assertion and reason, we will break down the problem step by step.

### Step 1: Understand the Assertion
The assertion states that when the height of a TV transmission tower is increased by three times, the range covered is doubled.

### Step 2: Understand the Reason
The reason given is that the range covered is proportional to the height of the TV transmission tower.

### Step 3: Use the Formula for Range
The range \( D \) of a TV transmission tower can be expressed using the formula:
\[ D = \sqrt{2hR} \]
where \( h \) is the height of the tower and \( R \) is a constant related to the Earth's radius.

### Step 4: Calculate the New Range
If the height \( h \) is increased by three times, the new height \( H \) becomes:
\[ H = 3h \]
Substituting this into the range formula gives:
\[ D' = \sqrt{2(3h)R} = \sqrt{6hR} \]

### Step 5: Compare the New Range to the Original Range
We can express the original range as:
\[ D = \sqrt{2hR} \]
Now, we can compare the new range \( D' \) with the original range \( D \):
\[ D' = \sqrt{6hR} = \sqrt{3} \cdot \sqrt{2hR} = \sqrt{3} D \]
Since \( \sqrt{3} \approx 1.732 \), we can see that \( D' \) is approximately 1.732 times the original range \( D \), not double.

### Step 6: Conclusion on the Assertion
Since the new range is approximately 1.732 times the original range, the assertion that the range is doubled is **false**.

### Step 7: Analyze the Reason
The reason states that the range is proportional to the height of the TV transmission tower. However, from the formula \( D \propto \sqrt{h} \), we see that the range is actually proportional to the square root of the height, not the height itself. Therefore, the reason is also **false**.

### Final Conclusion
- Assertion: **False**
- Reason: **False**