A metal plate of area `1.2 m^(2)` increases its area by `2.4 xx 10^(-4) m^(2)` when it is heated from `0^(@)C` to `100^(@) C`. The coefficient of cubical excpansion of the metal expressed in per `.^(@)C` is

Options

A. `2 xx 10^(-6)`
B. `4 xx 10^(-6)`
C. `6 xx 10^(-6)`
D. `3 xx 10^(-6)`

Explanation

To find the coefficient of cubical expansion of the metal, we can follow these steps:

### Step 1: Understand the relationship between area expansion and temperature change.
The change in area (\( \Delta A \)) of a plate due to temperature change is given by the formula:
\[ \Delta A = A_1 \cdot \beta \cdot \Delta T \]
where:
- \( \Delta A \) = change in area
- \( A_1 \) = initial area
- \( \beta \) = coefficient of areal expansion
- \( \Delta T \) = change in temperature

### Step 2: Rearrange the formula to find the coefficient of areal expansion (\( \beta \)).
Rearranging the formula gives:
\[ \beta = \frac{\Delta A}{A_1 \cdot \Delta T} \]

### Step 3: Substitute the known values into the equation.
We know:
- \( \Delta A = 2.4 \times 10^{-4} \, m^2 \)
- \( A_1 = 1.2 \, m^2 \)
- \( \Delta T = T_2 - T_1 = 100^\circ C - 0^\circ C = 100^\circ C \)

Substituting these values into the equation:
\[ \beta = \frac{2.4 \times 10^{-4} \, m^2}{1.2 \, m^2 \cdot 100^\circ C} \]

### Step 4: Calculate \( \beta \).
Calculating the right-hand side:
\[ \beta = \frac{2.4 \times 10^{-4}}{1.2 \times 100} = \frac{2.4 \times 10^{-4}}{120} = 2.0 \times 10^{-6} \, \text{per } ^\circ C \]

### Step 5: Relate the coefficient of areal expansion (\( \beta \)) to the coefficient of cubical expansion (\( \gamma \)).
The relationship between the coefficients is given by:
\[ \gamma = \frac{3}{2} \cdot \beta \]

### Step 6: Substitute \( \beta \) into the equation for \( \gamma \).
Substituting the value of \( \beta \):
\[ \gamma = \frac{3}{2} \cdot (2.0 \times 10^{-6}) = 3.0 \times 10^{-6} \, \text{per } ^\circ C \]

### Final Answer:
The coefficient of cubical expansion of the metal is:
\[ \gamma = 3.0 \times 10^{-6} \, \text{per } ^\circ C \]

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