$ ext{If } \alpha, \beta, ext{ and } \gamma ext{ are the coefficients of linear, area and volume expansion of the solid cube, the relation between } \alpha, \beta ext{ and } \gamma ext{ is :}$

Q: $\text{If } \alpha, \beta, \text{ and } \gamma \text{ are the coefficients of linear, area and volume expansion of the solid cube, the relation between } \alpha, \beta \text{ and } \gamma \text{ is :}$

$\text{If } \alpha, \beta, \text{ and } \gamma \text{ are the coefficients of linear, area and volume expansion of the solid cube, the relation between } \alpha, \beta \text{ and } \gamma \text{ is :}$

Question

$	ext{If } \alpha, \beta, 	ext{ and } \gamma 	ext{ are the coefficients of linear, area and volume expansion of the solid cube, the relation between } \alpha, \beta 	ext{ and } \gamma 	ext{ is :}$

Accepted Answer

To understand why Option 1 is the correct answer, we need to delve into the concepts of linear, area, and volume expansion coefficients for a solid cube.

### Explanation:

1. **Linear Expansion Coefficient ($\alpha$)**:
   - This coefficient represents the fractional change in length per degree change in temperature. For a solid cube, if the original length of a side is $L$, the change in length $\Delta L$ due to a temperature change $\Delta T$ is given by:
     $$
     \Delta L = \alpha L \Delta T
     $$

2. **Area Expansion Coefficient ($\beta$)**:
   - This coefficient represents the fractional change in area per degree change in temperature. For a face of the cube with area $A = L^2$, the change in area $\Delta A$ is:
     $$
     \Delta A = \beta A \Delta T
     $$
   - Since area is a two-dimensional measure, the area expansion coefficient is related to the linear expansion coefficient by:
     $$
     \beta = 2\alpha
     $$

3. **Volume Expansion Coefficient ($\gamma$)**:
   - This coefficient represents the fractional change in volume per degree change in temperature. For the cube with volume $V = L^3$, the change in volume $\Delta V$ is:
     $$
     \Delta V = \gamma V \Delta T
     $$
   - Since volume is a three-dimensional measure, the volume expansion coefficient is related to the linear expansion coefficient by:
     $$
     \gamma = 3\alpha
     $$

### Justification for Option 1:

Given the relationships:
- $\beta = 2\alpha$
- $\gamma = 3\alpha$

Option 1 states: $6\alpha = 3\beta = 2\gamma$.

Let's verify this:
- From $\beta = 2\alpha$, we have $3\beta = 3(2\alpha) = 6\alpha$.
- From $\gamma = 3\alpha$, we have $2\gamma = 2(3\alpha) = 6\alpha$.

Thus, $6\alpha = 3\beta = 2\gamma$ holds true, confirming that Option 1 is correct.

### Explanation for Incorrectness of Other Options:

- **Option 2: $3\alpha = 6\beta = \gamma$**
  - This implies $\beta = \frac{\gamma}{6}$ and $\alpha = \frac{\gamma}{3}$. However, we know $\gamma = 3\alpha$, which contradicts $\gamma = 6\beta$ since $\beta = 2\alpha$.

- **Option 3: $6\alpha = 2\beta = 3\gamma$**
  - This implies $\beta = 3\alpha$ and $\gamma = 2\alpha$, which contradicts the established relationships $\beta = 2\alpha$ and $\gamma = 3\alpha$.

- **Option 4: $3\alpha = \beta = 2\gamma$**
  - This implies $\beta = 3\alpha$ and $\gamma = \frac{\beta}{2}$, which contradicts $\beta = 2\alpha$ and $\gamma = 3\alpha$.

In conclusion, Option 1 is the only option that correctly represents the relationship between the coefficients of linear, area, and volume expansion for a solid cube.