Choose the correct statement:

Options

A. `alpha:beta:gamma::1:3:2`
B. `alpha:beta:gamma::3:2:1`
C. `alpha:beta:gamma::2:3:1`
D. `alpha:beta:gamma::1:2:3`

Explanation

To solve the question regarding the relationship between the coefficients of linear expansion (α), superficial expansion (β), and cubical expansion (γ), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Coefficients**:
- **Coefficient of Linear Expansion (α)**: This coefficient measures how much a material expands per unit length for a unit change in temperature. The formula is:
\[ \Delta L = \alpha \cdot L \cdot \Delta T \]
- **Coefficient of Superficial Expansion (β)**: This coefficient measures how much a surface area expands for a unit change in temperature. The formula is:
\[ \Delta A = \beta \cdot A \cdot \Delta T \]
- **Coefficient of Cubical Expansion (γ)**: This coefficient measures how much a volume expands for a unit change in temperature. The formula is:
\[ \Delta V = \gamma \cdot V \cdot \Delta T \]

2. **Relating the Coefficients**:
- For a linear dimension (length), the change in length is proportional to α.
- For a two-dimensional surface (area), if we consider a square with side length L, the area A = L². The change in area will be:
\[ \Delta A = 2\alpha \cdot L^2 \cdot \Delta T \implies \beta = 2\alpha \]
- For a three-dimensional object (volume), if we consider a cube with side length L, the volume V = L³. The change in volume will be:
\[ \Delta V = 3\alpha \cdot L^3 \cdot \Delta T \implies \gamma = 3\alpha \]

3. **Establishing the Ratios**:
- From the relationships derived:
\[ \beta = 2\alpha \quad \text{and} \quad \gamma = 3\alpha \]
- Therefore, the ratios of the coefficients can be expressed as:
\[ \alpha : \beta : \gamma = \alpha : 2\alpha : 3\alpha \]
- Simplifying this gives:
\[ 1 : 2 : 3 \]

4. **Choosing the Correct Statement**:
- The correct statement regarding the relationship between α, β, and γ is:
\[ \alpha : \beta : \gamma = 1 : 2 : 3 \]
- Thus, the correct answer is option D: 1 is to 2 is to 3.