Ratio of amplitude for two wave is 1:6.Find the ratio of intensity?

Answer

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

### Step-by-Step Solution:

1. **Identify the Given Data:**
- The ratio of amplitudes of two waves is given as \( A_1 : A_2 = 1 : 6 \).
- This can be expressed as \( \frac{A_1}{A_2} = \frac{1}{6} \).

2. **Understand the Relationship Between Intensity and Amplitude:**
- The intensity \( I \) of a wave is proportional to the square of its amplitude. This can be mathematically represented as:
\[ I \propto A^2 \]
- Therefore, the ratio of intensities \( \frac{I_1}{I_2} \) can be expressed in terms of the ratio of amplitudes:
\[ \frac{I_1}{I_2} = \left(\frac{A_1}{A_2}\right)^2 \]

3. **Substitute the Known Values:**
- We already have \( \frac{A_1}{A_2} = \frac{1}{6} \). Now, we substitute this value into the intensity ratio formula:
\[ \frac{I_1}{I_2} = \left(\frac{1}{6}\right)^2 \]

4. **Calculate the Ratio of Intensities:**
- Now, we calculate \( \left(\frac{1}{6}\right)^2 \):
\[ \frac{I_1}{I_2} = \frac{1^2}{6^2} = \frac{1}{36} \]

5. **Final Result:**
- The ratio of intensities \( I_1 : I_2 \) is therefore \( 1 : 36 \).

### Conclusion:
The ratio of intensity for the two waves is \( 1 : 36 \).

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