The speed of sound in air at `15^(@)C` and 76 cm of Hg is 340 m/s. The speed of sound in air at `30^(@)C` and 75 cm of Hg will be in m/s)

Options

A. `340 sqrt((303)/(288))`
B. `340 sqrt((208)/(303))`
C. `340sqrt(2)`
D. `340 sqrt((2xx75)/(76))`

Answer

To solve the problem, we need to find the speed of sound in air at a temperature of 30°C and a pressure of 75 cm of Hg, given that the speed of sound at 15°C and 76 cm of Hg is 340 m/s.

### Step-by-Step Solution:

1. **Understand the relationship between speed of sound and temperature**:
The speed of sound in air is directly proportional to the square root of the absolute temperature (in Kelvin). This can be expressed as:
\[ V \propto \sqrt{T} \]
Therefore, we can write:
\[ \frac{V_1}{V_2} = \sqrt{\frac{T_1}{T_2}} \]

2. **Convert temperatures from Celsius to Kelvin**:
- For \( T_1 = 15^\circ C \):
\[ T_1 = 15 + 273 = 288 \, K \]
- For \( T_2 = 30^\circ C \):
\[ T_2 = 30 + 273 = 303 \, K \]

3. **Substitute the known values into the equation**:
We know \( V_1 = 340 \, m/s \), \( T_1 = 288 \, K \), and \( T_2 = 303 \, K \). Substituting these values into the equation gives:
\[ \frac{340}{V_2} = \sqrt{\frac{288}{303}} \]

4. **Cross-multiply to solve for \( V_2 \)**:
Rearranging the equation, we get:
\[ V_2 = 340 \cdot \sqrt{\frac{303}{288}} \]

5. **Calculate the value of \( V_2 \)**:
First, calculate \( \frac{303}{288} \):
\[ \frac{303}{288} \approx 1.05208 \]
Now, take the square root:
\[ \sqrt{1.05208} \approx 1.0257 \]
Now, multiply by 340:
\[ V_2 \approx 340 \cdot 1.0257 \approx 348.4 \, m/s \]

6. **Final Answer**:
The speed of sound in air at 30°C and 75 cm of Hg is approximately \( 348.4 \, m/s \).