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) A and R both are true and R is correct explaination of A
B. b) A is true but R is false
C. c) A is false but R is true
D. d) Both A and R are false

Answer

To solve the question, we need to analyze the assertion and reason provided regarding the relationship between the height of a TV transmission tower and the range it covers.

### Step-by-Step Solution:

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

2. **Understanding the Reason**:
- The reason states that the range covered is proportional to the height of the TV transmission tower.

3. **Using the Range Formula**:
- The range \( R \) of a TV transmission tower can be expressed using the formula:
\[ R = \sqrt{2hR_e} \]
where \( h \) is the height of the tower and \( R_e \) is the radius of the Earth (which is a constant).

4. **Analyzing the Relationship**:
- From the formula, we can see that the range \( R \) is proportional to the square root of the height \( h \):
\[ R \propto \sqrt{h} \]
- This means that if we increase the height of the tower, the range will increase with the square root of that height.

5. **Calculating the New Range**:
- Let the initial height of the tower be \( h_1 \) and the initial range be \( R_1 \).
- If the height is increased to \( h_2 = 3h_1 \), the new range \( R_2 \) can be expressed as:
\[ R_2 = \sqrt{2(3h_1)R_e} = \sqrt{3} \cdot \sqrt{2h_1R_e} = \sqrt{3} R_1 \]
- Since \( \sqrt{3} \) is approximately 1.732, this means \( R_2 \) is not double \( R_1 \) but rather about 1.732 times \( R_1 \).

6. **Conclusion**:
- Therefore, the assertion that the range is doubled when the height is tripled is false.
- The reason that the range is proportional to the height is also misleading because it is actually proportional to the square root of the height.

7. **Final Answer**:
- Both the assertion (A) and the reason (R) are false. Hence, the correct option is (d): both A and R are false.