In a radar system, peak transmitted power is increased a factor of 81 and the antenna diameter is increased by a factor of 3, then the maximum-range will increase by a factor of

Options

A. 81
B. 27
C. 9
D. 3

Explanation

To solve the problem, we need to determine how the maximum range \( R_{mx} \) of a radar system changes when the peak transmitted power and the antenna diameter are increased.

### Step-by-Step Solution:

1. **Understanding the Relationship**:
The maximum range \( R_{mx} \) of a radar system is directly proportional to the product of the antenna diameter \( D \) and the fourth root of the transmitted power \( P \). This can be expressed mathematically as:
\[ R_{mx} \propto D \cdot P^{1/4} \]

2. **Identifying Initial and Final Values**:
Let:
- Initial power \( P_1 = P \)
- Final power \( P_2 = 81P \) (since the power is increased by a factor of 81)
- Initial diameter \( D_1 = D \)
- Final diameter \( D_2 = 3D \) (since the diameter is increased by a factor of 3)

3. **Setting Up the Ratio of Maximum Ranges**:
We can express the ratio of the final maximum range \( R_2 \) to the initial maximum range \( R_1 \):
\[ \frac{R_2}{R_1} = \frac{D_2}{D_1} \cdot \left(\frac{P_2}{P_1}\right)^{1/4} \]

4. **Substituting the Values**:
Substitute \( D_2 \), \( D_1 \), \( P_2 \), and \( P_1 \) into the equation:
\[ \frac{R_2}{R_1} = \frac{3D}{D} \cdot \left(\frac{81P}{P}\right)^{1/4} \]

5. **Simplifying the Expression**:
This simplifies to:
\[ \frac{R_2}{R_1} = 3 \cdot (81)^{1/4} \]
Now, calculate \( (81)^{1/4} \):
\[ (81)^{1/4} = (3^4)^{1/4} = 3 \]
Therefore:
\[ \frac{R_2}{R_1} = 3 \cdot 3 = 9 \]

6. **Conclusion**:
The maximum range \( R_{mx} \) will increase by a factor of 9.

### Final Answer:
The maximum range will increase by a factor of **9**.