If the ratio of amplitude of two waves is `4:3`, then the ratio of maximum and minimum intensity is

Options

A. (a) `16:18`
B. (b) `18:16`
C. (c) `49:1`
D. (d) `94:1`

Explanation

To solve the problem of finding the ratio of maximum and minimum intensity given the ratio of amplitudes of two waves as \(4:3\), we can follow these steps:

### Step 1: Understand the relationship between intensity and amplitude
The intensity \(I\) of a wave is proportional to the square of its amplitude \(A\). Therefore, we can express the intensities of two waves as:
\[ I_1 \propto A_1^2 \quad \text{and} \quad I_2 \propto A_2^2 \]

### Step 2: Define the amplitudes based on the given ratio
Let the amplitudes of the two waves be:
\[ A_1 = 4k \quad \text{and} \quad A_2 = 3k \]
where \(k\) is a constant.

### Step 3: Calculate the intensities using the amplitudes
Using the relationship between intensity and amplitude:
\[ I_1 = A_1^2 = (4k)^2 = 16k^2 \]
\[ I_2 = A_2^2 = (3k)^2 = 9k^2 \]

### Step 4: Write the expressions for maximum and minimum intensity
The maximum intensity \(I_{\text{max}}\) and minimum intensity \(I_{\text{min}}\) can be expressed as:
\[ I_{\text{max}} = I_1 + I_2 + 2\sqrt{I_1 I_2} \]
\[ I_{\text{min}} = I_1 + I_2 - 2\sqrt{I_1 I_2} \]

### Step 5: Substitute the values of \(I_1\) and \(I_2\)
Substituting \(I_1\) and \(I_2\) into the equations:
\[ I_{\text{max}} = 16k^2 + 9k^2 + 2\sqrt{16k^2 \cdot 9k^2} \]
\[ I_{\text{min}} = 16k^2 + 9k^2 - 2\sqrt{16k^2 \cdot 9k^2} \]

### Step 6: Simplify the expressions
Calculating the square root:
\[ \sqrt{16k^2 \cdot 9k^2} = \sqrt{144k^4} = 12k^2 \]
Thus, we can rewrite \(I_{\text{max}}\) and \(I_{\text{min}}\):
\[ I_{\text{max}} = 25k^2 + 24k^2 = 49k^2 \]
\[ I_{\text{min}} = 25k^2 - 24k^2 = k^2 \]

### Step 7: Find the ratio of maximum to minimum intensity
Now, we can find the ratio:
\[ \frac{I_{\text{max}}}{I_{\text{min}}} = \frac{49k^2}{k^2} = 49 \]

### Conclusion
The ratio of maximum intensity to minimum intensity is:
\[ \boxed{49:1} \]