If a cylinder of diameter `1.0cm` at `30^(@)C` is to be slid into a hole of diameter `0.9997 cm` in a steel plate at the same temperature, the minimum required rise in the temperature of the plate is: (Coefficient of linear expansion of steel`= 12 xx 10^(-6//@)C`)

Options

A. `25^(@)C`
B. `35^(@)C`
C. `45^(@)C`
D. `55^(@)C`

Explanation

To solve the problem, we need to determine the minimum required rise in temperature of the steel plate so that the diameter of the hole expands enough to accommodate the cylinder.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Diameter of the cylinder, \( D_c = 1.0 \, \text{cm} = 0.01 \, \text{m} \)
- Diameter of the hole, \( D_h = 0.9997 \, \text{cm} = 0.009997 \, \text{m} \)
- Initial temperature, \( T_1 = 30^\circ C \)
- Coefficient of linear expansion of steel, \( \alpha = 12 \times 10^{-6} \, \text{°C}^{-1} \)

2. **Determine the Required Expansion:**
- The hole needs to expand from \( D_h \) to \( D_c \).
- The required expansion in diameter, \( \Delta D = D_c - D_h = 0.01 \, \text{m} - 0.009997 \, \text{m} = 0.0003 \, \text{m} \).

3. **Use the Linear Expansion Formula:**
- The formula for linear expansion is given by:
\[ L_2 = L_1 (1 + \alpha T) \]
- Here, \( L_1 \) is the initial diameter of the hole, and \( L_2 \) is the final diameter after expansion. We can set \( L_2 = D_c \) and \( L_1 = D_h \).

4. **Set Up the Equation:**
- Plugging in the values:
\[ D_c = D_h (1 + \alpha T) \]
- Rearranging gives:
\[ 1 = \frac{D_h}{D_c} (1 + \alpha T) \]

5. **Substituting Values:**
- Substitute \( D_h = 0.009997 \, \text{m} \) and \( D_c = 0.01 \, \text{m} \):
\[ 1 = \frac{0.009997}{0.01} (1 + 12 \times 10^{-6} T) \]
- This simplifies to:
\[ 1 = 0.9997 (1 + 12 \times 10^{-6} T) \]

6. **Solving for \( T \):**
- Expanding the equation:
\[ 1 = 0.9997 + 0.9997 \times 12 \times 10^{-6} T \]
- Rearranging gives:
\[ 1 - 0.9997 = 0.9997 \times 12 \times 10^{-6} T \]
- Thus:
\[ 0.0003 = 0.9997 \times 12 \times 10^{-6} T \]
- Solving for \( T \):
\[ T = \frac{0.0003}{0.9997 \times 12 \times 10^{-6}} \approx 25.0 \, \text{°C} \]

### Conclusion:
The minimum required rise in the temperature of the plate is approximately \( 25.0 \, \text{°C} \).