Find the temperature at which the velocity of sound in air will be `1 1/2` times the velocity at `11^@ C`.

Answer

To find the temperature at which the velocity of sound in air is \(1.5\) times the velocity at \(11^\circ C\), we can follow these steps:

### Step 1: Understand the relationship between temperature and the velocity of sound
The velocity of sound in air can be approximated by the formula:
\[ v = v_0 \sqrt{\frac{T}{T_0}} \]
where:
- \(v\) is the velocity of sound at temperature \(T\),
- \(v_0\) is the velocity of sound at a reference temperature \(T_0\),
- \(T\) is the absolute temperature in Kelvin,
- \(T_0\) is the absolute temperature in Kelvin at which \(v_0\) is measured.

### Step 2: Convert the reference temperature to Kelvin
The reference temperature \(T_0\) is given as \(11^\circ C\). To convert this to Kelvin:
\[ T_0 = 11 + 273.15 = 284.15 \, K \]

### Step 3: Calculate the velocity of sound at \(11^\circ C\)
The velocity of sound in air at \(11^\circ C\) is approximately \(v_0 = 343 \, m/s\) (this is a standard value).

### Step 4: Set up the equation for the new velocity
We want to find the temperature \(T\) at which the velocity of sound is \(1.5\) times the velocity at \(11^\circ C\):
\[ v = 1.5 \times v_0 = 1.5 \times 343 \, m/s = 514.5 \, m/s \]

### Step 5: Set up the equation using the velocity formula
Using the velocity formula:
\[ 514.5 = 343 \sqrt{\frac{T}{284.15}} \]

### Step 6: Solve for \(T\)
First, divide both sides by \(343\):
\[ \sqrt{\frac{T}{284.15}} = \frac{514.5}{343} \]
Now, square both sides:
\[ \frac{T}{284.15} = \left(\frac{514.5}{343}\right)^2 \]
Calculating the right side:
\[ \frac{514.5}{343} \approx 1.5 \quad \text{(since \(514.5\) is \(1.5\) times \(343\))} \]
Thus:
\[ \frac{T}{284.15} = (1.5)^2 = 2.25 \]
Now, multiply both sides by \(284.15\):
\[ T = 2.25 \times 284.15 \approx 639.34 \, K \]

### Step 7: Convert \(T\) back to Celsius
To convert back to Celsius:
\[ T_{Celsius} = T - 273.15 \approx 639.34 - 273.15 \approx 366.19^\circ C \]

### Final Answer
The temperature at which the velocity of sound in air will be \(1.5\) times the velocity at \(11^\circ C\) is approximately \(366.19^\circ C\).

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