Driver of a truck gets his steel petrol tank filled with `75 L` of petrol at `10^(@)C`. If a `alpha_("steel")` is `12 xx 10^(-6)//.^(@)C` and `gamma_("pet")` is `9.5 xx 10^(-4)//.^(@)C` the overflow of petrol at `30^(2)C` is -

Options

A. `7.31 L`
B. `1.37 L`
C. `13.7 L`
D. `1.73 L`

Explanation

To solve the problem, we need to calculate the overflow of petrol when the temperature changes from \(10^\circ C\) to \(30^\circ C\). We will use the coefficients of linear and volume expansion provided.

### Step 1: Write down the given data
- Volume of petrol, \(V_1 = 75 \, L\)
- Initial temperature, \(T_1 = 10^\circ C\)
- Final temperature, \(T_2 = 30^\circ C\)
- Coefficient of linear expansion of steel, \(\alpha_{steel} = 12 \times 10^{-6} \, /^{\circ}C\)
- Coefficient of volume expansion of petrol, \(\gamma_{petrol} = 9.5 \times 10^{-4} \, /^{\circ}C\)

### Step 2: Calculate the volume expansion coefficient of steel
The volume expansion coefficient of steel (\(\gamma_{steel}\)) can be calculated using the relation:
\[ \gamma_{steel} = 3 \times \alpha_{steel} \]
Substituting the value:
\[ \gamma_{steel} = 3 \times (12 \times 10^{-6}) = 36 \times 10^{-6} \, /^{\circ}C \]

### Step 3: Calculate the effective volume expansion coefficient
The effective volume expansion coefficient for the overflow (\(\gamma_{overflow}\)) is given by:
\[ \gamma_{overflow} = \gamma_{petrol} - \gamma_{steel} \]
Substituting the values:
\[ \gamma_{overflow} = (9.5 \times 10^{-4}) - (36 \times 10^{-6}) = (950 \times 10^{-6}) - (36 \times 10^{-6}) = 914 \times 10^{-6} \, /^{\circ}C \]

### Step 4: Calculate the change in temperature
The change in temperature (\(\Delta T\)) is:
\[ \Delta T = T_2 - T_1 = 30^\circ C - 10^\circ C = 20^\circ C \]

### Step 5: Calculate the change in volume of petrol
Using the formula for volume expansion:
\[ \Delta V = V_1 \cdot \gamma_{overflow} \cdot \Delta T \]
Substituting the values:
\[ \Delta V = 75 \, L \cdot (914 \times 10^{-6}) \cdot (20) \]
Calculating this:
\[ \Delta V = 75 \cdot 914 \cdot 20 \times 10^{-6} = 1371 \times 10^{-6} \, L = 1.371 \, L \]

### Step 6: Final answer
The overflow of petrol at \(30^\circ C\) is approximately:
\[ \Delta V \approx 1.37 \, L \]