A rectangular block is heated from `0^(@)C` to `100^(@)C`. The percentage increases in its length is 0.10 % what will be the percentage increases in it volume ?

Options

A. `0.03%`
B. `0.10%`
C. `0.30%`
D. none of these

Explanation

To solve the problem of finding the percentage increase in the volume of a rectangular block when it is heated from \(0^\circ C\) to \(100^\circ C\), given that the percentage increase in its length is \(0.10\%\), we can use the relationship between linear expansion and volumetric expansion.

### Step-by-Step Solution:

1. **Understand the relationship between linear and volumetric expansion**:
- The coefficient of linear expansion (\(\alpha\)) relates to the change in length per unit length per degree change in temperature.
- The coefficient of volumetric expansion (\(\gamma\)) relates to the change in volume per unit volume per degree change in temperature.
- The relationship between these coefficients is given by:
\[ \gamma = 3\alpha \]

2. **Given data**:
- Percentage increase in length (\(\Delta L/L\)) = \(0.10\%\) = \(0.001\) (as a decimal).

3. **Calculate the percentage increase in volume**:
- Since \(\gamma = 3\alpha\), we can express the percentage increase in volume as:
\[ \frac{\Delta V}{V} = \gamma \Delta T \]
- Using the relationship, we can substitute \(\gamma\) with \(3\alpha\):
\[ \frac{\Delta V}{V} = 3 \left(\frac{\Delta L}{L}\right) \]
- Now substituting the given percentage increase in length:
\[ \frac{\Delta V}{V} = 3 \times 0.001 = 0.003 \]

4. **Convert the decimal back to percentage**:
- To convert \(0.003\) to percentage:
\[ \Delta V/V = 0.003 \times 100\% = 0.30\% \]

5. **Conclusion**:
- The percentage increase in volume of the rectangular block when heated from \(0^\circ C\) to \(100^\circ C\) is \(0.30\%\).

### Final Answer:
The percentage increase in volume is **0.30%**.

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