Imagine a hypothetical material whose average temperature coefficient of linear expansion is `0.1//.^@C` .Find the fractional increase in area of thin square plate of above material when temperature is increase by `10^@C`

Answer

To solve the problem of finding the fractional increase in the area of a thin square plate made of a hypothetical material with a given temperature coefficient of linear expansion, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values**:
- Coefficient of linear expansion (α) = 0.1 /°C
- Change in temperature (ΔT) = 10 °C

2. **Understand the Relationship**:
- The fractional change in area (ΔA/A) can be related to the coefficient of linear expansion. The formula for the change in area due to temperature change is:
\[ \frac{\Delta A}{A} = \beta \cdot \Delta T \]
- Where β is the coefficient of area expansion.

3. **Calculate the Coefficient of Area Expansion (β)**:
- The coefficient of area expansion (β) is related to the coefficient of linear expansion (α) by the formula:
\[ \beta = 2 \cdot \alpha \]
- Substituting the value of α:
\[ \beta = 2 \cdot 0.1 = 0.2 \, /°C \]

4. **Substitute Values into the Area Change Formula**:
- Now, substitute β and ΔT into the area change formula:
\[ \frac{\Delta A}{A} = 0.2 \cdot 10 \]

5. **Calculate the Fractional Increase in Area**:
- Perform the multiplication:
\[ \frac{\Delta A}{A} = 2 \]

### Final Answer:
The fractional increase in area of the thin square plate when the temperature is increased by 10 °C is **2**.

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