At what temperature will the speed of sound in air be 3 times its value at `0^(@)`C?

Options

A. `1184^(@)C`
B. `1148^(@)C`
C. `2184^(@)C`
D. `2148^(@)C`

Explanation

To find the temperature at which the speed of sound in air is three times its value at 0°C, we can follow these steps:

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

### Step 2: Establish the values at 0°C
At 0°C, the absolute temperature \( T_0 \) in Kelvin is:
\[ T_0 = 0 + 273 = 273 \, K \]
Let the speed of sound at 0°C be \( V_0 \).

### Step 3: Set up the equation for the desired speed
We want to find the temperature \( T \) at which the speed of sound \( V \) is three times \( V_0 \):
\[ V = 3V_0 \]

### Step 4: Use the proportionality to set up the equation
Using the proportionality of speed to the square root of temperature, we have:
\[ V_0 \propto \sqrt{T_0} \]
\[ 3V_0 \propto \sqrt{T} \]

### Step 5: Set up the ratio
From the proportionality, we can write:
\[ \frac{3V_0}{V_0} = \frac{\sqrt{T}}{\sqrt{T_0}} \]
This simplifies to:
\[ 3 = \frac{\sqrt{T}}{\sqrt{273}} \]

### Step 6: Square both sides to eliminate the square root
Squaring both sides gives:
\[ 9 = \frac{T}{273} \]

### Step 7: Solve for \( T \)
Multiplying both sides by 273 gives:
\[ T = 9 \times 273 \]
\[ T = 2457 \, K \]

### Step 8: Convert back to Celsius
To convert the temperature back to Celsius:
\[ T_{Celsius} = T - 273 \]
\[ T_{Celsius} = 2457 - 273 = 2184 \, °C \]

### Final Answer
The temperature at which the speed of sound in air is three times its value at 0°C is:
\[ T = 2184 \, °C \]

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