A cube at temperature `0^(@)` C is compressed equal from all sides by an external pressure P. BY what amount should its temperature be raised to bring it back to the size it had before the external pressure was applied. The bulk modulus of the material of the cube is K and the coefficient of linear expansion is `alpha`.

Options

A. `P//K alpha `
B. `P//3 K alpha `
C. `3pi alpha//K`
D. `3K//P `

Explanation

To solve the problem, we need to determine the amount by which the temperature of a cube must be raised to restore its original size after being compressed by an external pressure \( P \). We will use the concepts of bulk modulus and thermal expansion in our calculations.

### Step-by-Step Solution:

1. **Understanding Bulk Modulus**:
The bulk modulus \( K \) is defined as:
\[ K = -\frac{\Delta P}{\frac{\Delta V}{V_0}} \]
where \( \Delta P \) is the change in pressure, \( \Delta V \) is the change in volume, and \( V_0 \) is the original volume.

2. **Relating Volume Change to Temperature Change**:
The change in volume \( \Delta V \) due to a change in temperature \( \Delta T \) can be expressed using the volumetric thermal expansion coefficient \( \gamma \):
\[ \Delta V = V_0 \cdot \gamma \cdot \Delta T \]
For a cube, the volumetric expansion coefficient \( \gamma \) is related to the linear expansion coefficient \( \alpha \) by:
\[ \gamma = 3\alpha \]
Therefore, we can rewrite the volume change as:
\[ \Delta V = V_0 \cdot 3\alpha \cdot \Delta T \]

3. **Substituting into the Bulk Modulus Equation**:
Substituting \( \Delta V \) into the bulk modulus equation gives us:
\[ K = -\frac{P}{\frac{3\alpha \Delta T}{V_0}} \]
Rearranging this, we find:
\[ \Delta T = -\frac{P \cdot V_0}{3K\alpha} \]

4. **Interpreting the Signs**:
Since the cube is compressed, \( \Delta V \) is negative, which means that the temperature must be increased (hence the negative sign in front of \( \Delta T \)) to restore the original volume. Thus, we take the absolute value:
\[ \Delta T = \frac{P \cdot V_0}{3K\alpha} \]

5. **Final Expression**:
Since the original volume \( V_0 \) is not specified, we can express the final result in terms of \( P \), \( K \), and \( \alpha \):
\[ \Delta T = \frac{P}{3K\alpha} \]

### Conclusion:
The amount by which the temperature should be raised to restore the cube to its original size is given by:
\[ \Delta T = \frac{P}{3K\alpha} \]