In the case of interference, the maximum and minimum intensities are in the ratio 16 : 9. Then

Options

A. The maximum and minimum amplitudes will be in the ratio 9 : 5
B. The intensities of the individual waves will be in the ratio 4 : 3
C. The amplitudes of the individual waves will be in the ratio 7 : 1
D. none of the above is true

Answer

To solve the problem, we need to analyze the given ratio of maximum and minimum intensities in the context of wave interference.

### Step-by-Step Solution:

1. **Understanding the Given Ratio**:
The problem states that the maximum intensity \( I_{\text{max}} \) and minimum intensity \( I_{\text{min}} \) are in the ratio of 16:9. This can be expressed mathematically as:
\[ \frac{I_{\text{max}}}{I_{\text{min}}} = \frac{16}{9} \]

2. **Relating Intensity to Amplitude**:
We know that intensity \( I \) is proportional to the square of the amplitude \( A \):
\[ I \propto A^2 \]
Therefore, we can write:
\[ \frac{I_{\text{max}}}{I_{\text{min}}} = \frac{A_{\text{max}}^2}{A_{\text{min}}^2} \]

3. **Setting Up the Equation**:
Substituting the ratio of intensities into the equation gives:
\[ \frac{A_{\text{max}}^2}{A_{\text{min}}^2} = \frac{16}{9} \]

4. **Taking the Square Root**:
To find the ratio of amplitudes, we take the square root of both sides:
\[ \frac{A_{\text{max}}}{A_{\text{min}}} = \frac{\sqrt{16}}{\sqrt{9}} = \frac{4}{3} \]

5. **Conclusion for Amplitude Ratio**:
Thus, the ratio of the maximum amplitude to the minimum amplitude is:
\[ A_{\text{max}} : A_{\text{min}} = 4 : 3 \]

6. **Finding the Ratio of Individual Intensities**:
The maximum intensity \( I_{\text{max}} \) and minimum intensity \( I_{\text{min}} \) can be expressed in terms of the individual intensities \( I_1 \) and \( I_2 \) of the two waves:
\[ I_{\text{max}} = (A_1 + A_2)^2 \quad \text{and} \quad I_{\text{min}} = (A_1 - A_2)^2 \]
Using the ratio we found:
\[ \frac{(A_1 + A_2)^2}{(A_1 - A_2)^2} = \frac{16}{9} \]
This leads us to:
\[ \frac{A_1 + A_2}{A_1 - A_2} = \frac{4}{3} \]

7. **Cross Multiplying**:
Cross-multiplying gives:
\[ 3(A_1 + A_2) = 4(A_1 - A_2) \]
Expanding both sides:
\[ 3A_1 + 3A_2 = 4A_1 - 4A_2 \]

8. **Rearranging Terms**:
Rearranging the equation leads to:
\[ 3A_2 + 4A_2 = 4A_1 - 3A_1 \]
Which simplifies to:
\[ 7A_2 = A_1 \]

9. **Finding the Ratio of Individual Intensities**:
Now substituting back into the intensity relation:
\[ \frac{I_1}{I_2} = \frac{A_1^2}{A_2^2} = \frac{(7A_2)^2}{A_2^2} = \frac{49}{1} \]

### Final Answer:
The correct ratios are:
- The ratio of maximum to minimum amplitude is \( 4:3 \).
- The ratio of individual intensities is \( 49:1 \).

Thus, the answer is that none of the given options are correct.