The highest frequency produced by a man is 1700 Hz and that of a woman is 2780 Hz . The ratio of wavelengths of sound of man and woman are (speed of sound is ` 340 m s^(-1)` )

Options

A. ` 1 : 0 . 34`
B. ` 1 : 0 . 61 `
C. ` 1 : 0 . 69`
D. ` 1 : 0 . 59`

Explanation

To find the ratio of the wavelengths of sound produced by a man and a woman, we can use the relationship between the speed of sound, frequency, and wavelength. The formula we will use is:

\[ v = f \cdot \lambda \]

Where:
- \( v \) is the speed of sound,
- \( f \) is the frequency,
- \( \lambda \) is the wavelength.

### Step-by-Step Solution:

1. **Identify the given values**:
- Speed of sound, \( v = 340 \, \text{m/s} \)
- Frequency of sound produced by a man, \( f_m = 1700 \, \text{Hz} \)
- Frequency of sound produced by a woman, \( f_w = 2780 \, \text{Hz} \)

2. **Write the wavelength formulas for both man and woman**:
- For the man:
\[ \lambda_m = \frac{v}{f_m} = \frac{340 \, \text{m/s}}{1700 \, \text{Hz}} \]
- For the woman:
\[ \lambda_w = \frac{v}{f_w} = \frac{340 \, \text{m/s}}{2780 \, \text{Hz}} \]

3. **Calculate the wavelengths**:
- Calculate \( \lambda_m \):
\[ \lambda_m = \frac{340}{1700} = 0.2 \, \text{m} \]
- Calculate \( \lambda_w \):
\[ \lambda_w = \frac{340}{2780} \approx 0.122 \, \text{m} \]

4. **Find the ratio of the wavelengths**:
\[ \text{Ratio} = \frac{\lambda_m}{\lambda_w} = \frac{0.2}{0.122} \approx 1.64 \]

5. **Express the ratio in simplest form**:
\[ \text{Ratio} = \frac{1.64}{1} \text{ or } \approx 1.64 : 1 \]

### Final Answer:
The ratio of the wavelengths of sound produced by a man and a woman is approximately \( 1.64 : 1 \).