An iron ball is heated. The percentage increase will be the largest in –

Options

A. diameter
B. surface area
C. volume
D. density

Answer

To solve the question regarding the percentage increase in the dimensions of an iron ball when heated, we will analyze the effects of thermal expansion on diameter, surface area, volume, and density step by step.

### Step-by-Step Solution:

1. **Understanding Thermal Expansion**:
- When an object is heated, it expands. This expansion can be categorized into three types: linear (length), area (surface), and volume.
- The coefficients of linear, area, and volume expansions are denoted as α, β, and γ respectively.

2. **Relationship Between Coefficients**:
- The relationships between these coefficients are given by:
- \( \beta = 2\alpha \) (for area expansion)
- \( \gamma = 3\alpha \) (for volume expansion)
- This means that the volume expansion is three times the linear expansion, and the area expansion is twice the linear expansion.

3. **Calculating Percentage Increase**:
- The percentage increase in length (diameter) can be expressed as:
\[ \text{Percentage Increase in Length} = \frac{\Delta L}{L} \times 100 = \alpha \Delta T \times 100 \]
- The percentage increase in area (surface area) can be expressed as:
\[ \text{Percentage Increase in Area} = \frac{\Delta A}{A} \times 100 = \beta \Delta T \times 100 = 2\alpha \Delta T \times 100 \]
- The percentage increase in volume can be expressed as:
\[ \text{Percentage Increase in Volume} = \frac{\Delta V}{V} \times 100 = \gamma \Delta T \times 100 = 3\alpha \Delta T \times 100 \]
- The density will decrease as the volume increases, so it is not applicable for percentage increase.

4. **Comparing the Increases**:
- From the equations derived:
- Percentage Increase in Length = \( \alpha \Delta T \times 100 \)
- Percentage Increase in Area = \( 2\alpha \Delta T \times 100 \)
- Percentage Increase in Volume = \( 3\alpha \Delta T \times 100 \)
- It is clear that the percentage increase in volume is the largest since it has the highest coefficient (3).

5. **Conclusion**:
- Therefore, the percentage increase will be the largest in **volume**.

### Final Answer:
The percentage increase will be the largest in **volume**.