Ratio of amplitude for two wave is 3:15 .Find the ratio of intensity?

Explanation

To solve the problem of finding the ratio of intensity given the ratio of amplitudes, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Ratio of Amplitudes**:
The problem states that the ratio of amplitudes for two waves is given as \( A_1 : A_2 = 3 : 15 \).

2. **Express the Ratio of Amplitudes**:
We can express this ratio in fraction form:
\[ \frac{A_1}{A_2} = \frac{3}{15} \]

3. **Simplify the Ratio of Amplitudes**:
Simplifying \( \frac{3}{15} \) gives:
\[ \frac{A_1}{A_2} = \frac{1}{5} \]

4. **Use the Relationship Between Intensity and Amplitude**:
The intensity \( I \) of a wave is directly proportional to the square of its amplitude. Therefore, we can write:
\[ \frac{I_1}{I_2} = \left(\frac{A_1}{A_2}\right)^2 \]

5. **Substitute the Ratio of Amplitudes into the Intensity Formula**:
Substitute \( \frac{A_1}{A_2} = \frac{1}{5} \) into the intensity ratio:
\[ \frac{I_1}{I_2} = \left(\frac{1}{5}\right)^2 \]

6. **Calculate the Square of the Ratio**:
Calculating the square gives:
\[ \frac{I_1}{I_2} = \frac{1}{25} \]

7. **Final Result**:
Thus, the ratio of intensities is:
\[ I_1 : I_2 = 1 : 25 \]