The velocity of sound in air at `0^(@)C` is 331 m/s. Find its velocity when temperature rises to `91^(@)C` and its pressure is doubled.

Options

A. 330 m/s
B. 165 m/s
C. `382.2m//s`
D. `332.9m//s`

Explanation

To find the velocity of sound in air at a temperature of \(91^\circ C\) when the pressure is doubled, we can follow these steps:

### Step 1: Understand the relationship between velocity of sound and temperature
The velocity of sound in air is given by the formula:
\[ V \propto \sqrt{T} \]
where \(T\) is the absolute temperature in Kelvin.

### Step 2: Convert temperatures to Kelvin
The initial temperature is \(0^\circ C\), which is:
\[ T_1 = 0 + 273 = 273 \, K \]
The final temperature is \(91^\circ C\), which is:
\[ T_2 = 91 + 273 = 364 \, K \]

### Step 3: Set up the ratio of velocities
Using the relationship between velocity and temperature, we can write:
\[ \frac{V_1}{V_2} = \sqrt{\frac{T_1}{T_2}} \]
where \(V_1\) is the initial velocity of sound and \(V_2\) is the final velocity of sound.

### Step 4: Substitute the known values
We know:
- \(V_1 = 331 \, m/s\)
- \(T_1 = 273 \, K\)
- \(T_2 = 364 \, K\)

Substituting these values into the ratio gives:
\[ \frac{331}{V_2} = \sqrt{\frac{273}{364}} \]

### Step 5: Solve for \(V_2\)
First, calculate the square root:
\[ \sqrt{\frac{273}{364}} = \sqrt{\frac{273}{364}} = \sqrt{0.750} \approx 0.866 \]

Now, rearranging the equation:
\[ V_2 = \frac{331}{0.866} \approx 382.2 \, m/s \]

### Step 6: Conclusion
The final velocity of sound in air at \(91^\circ C\) is approximately:
\[ V_2 \approx 382.2 \, m/s \]