What is the percentage increase in the speed of sound when temperature increases from -`5^@C` 10 `32^@C ?`

Answer

To find the percentage increase in the speed of sound when the temperature increases from -5°C to 32°C, we can follow these steps:

### Step 1: Convert temperatures from Celsius to Kelvin
The absolute temperature in Kelvin (K) is given by:
\[ T(K) = T(°C) + 273.15 \]

- For \( T_1 = -5°C \):
\[ T_1 = -5 + 273.15 = 268.15 \, K \]
- For \( T_2 = 32°C \):
\[ T_2 = 32 + 273.15 = 305.15 \, K \]

### Step 2: Use the relationship between speed of sound and temperature
The speed of sound \( V \) is directly proportional to the square root of the absolute temperature:
\[ V \propto \sqrt{T} \]

Thus, we can express the speeds at the two temperatures as:
\[ V_1 = k \sqrt{T_1} \]
\[ V_2 = k \sqrt{T_2} \]
where \( k \) is a constant.

### Step 3: Calculate the speeds at the two temperatures
Using the values of \( T_1 \) and \( T_2 \):
- Calculate \( V_1 \):
\[ V_1 = k \sqrt{268.15} \]
- Calculate \( V_2 \):
\[ V_2 = k \sqrt{305.15} \]

### Step 4: Find the percentage increase in speed
The percentage increase in speed is given by:
\[ \text{Percentage Increase} = \frac{V_2 - V_1}{V_1} \times 100 \]

Substituting the expressions for \( V_1 \) and \( V_2 \):
\[ \text{Percentage Increase} = \frac{k \sqrt{305.15} - k \sqrt{268.15}}{k \sqrt{268.15}} \times 100 \]

The \( k \) cancels out:
\[ \text{Percentage Increase} = \frac{\sqrt{305.15} - \sqrt{268.15}}{\sqrt{268.15}} \times 100 \]

### Step 5: Calculate the numerical values
Now we can calculate the numerical values:
1. Calculate \( \sqrt{305.15} \) and \( \sqrt{268.15} \):
- \( \sqrt{305.15} \approx 17.46 \)
- \( \sqrt{268.15} \approx 16.37 \)

2. Substitute these values back into the percentage increase formula:
\[ \text{Percentage Increase} = \frac{17.46 - 16.37}{16.37} \times 100 \]
\[ \text{Percentage Increase} = \frac{1.09}{16.37} \times 100 \approx 6.67\% \]

### Final Answer
The percentage increase in the speed of sound when the temperature increases from -5°C to 32°C is approximately **6.67%**.

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