The temperature of a piece of metal is raised from `27^(@)C` to `51.2^(@)C`. The rate at which the metal radiates energy increases nearly,

Options

A. 2 times
B. 4 times
C. `4.46 times`
D. `1.36 times`

Explanation

To solve the problem of how much the rate at which the metal radiates energy increases when its temperature is raised from 27°C to 51.2°C, we can use the Stefan-Boltzmann Law. This law states that the power radiated by a black body is proportional to the fourth power of its absolute temperature (in Kelvin).

### Step-by-Step Solution:

1. **Convert the temperatures from Celsius to Kelvin:**
- Initial temperature, \( T_i = 27°C + 273 = 300 K \)
- Final temperature, \( T_f = 51.2°C + 273 = 324.2 K \)

2. **Apply the Stefan-Boltzmann Law:**
- The power radiated (E) is proportional to \( T^4 \):
\[ E \propto T^4 \]
- Therefore, we can express the initial and final energy radiated as:
\[ E_i = k \cdot (T_i)^4 = k \cdot (300)^4 \]
\[ E_f = k \cdot (T_f)^4 = k \cdot (324.2)^4 \]
where \( k \) is the Stefan-Boltzmann constant.

3. **Calculate \( E_i \) and \( E_f \):**
- Calculate \( E_i \):
\[ E_i = k \cdot (300)^4 = k \cdot 81 \times 10^8 \]
- Calculate \( E_f \):
\[ E_f = k \cdot (324.2)^4 \approx k \cdot 110.2 \times 10^8 \]

4. **Find the change in energy radiated:**
- Change in energy, \( \Delta E = E_f - E_i \):
\[ \Delta E = (110.2 \times 10^8) - (81 \times 10^8) = 29.2 \times 10^8 \]

5. **Calculate the percentage increase in energy radiated:**
- Percentage increase formula:
\[ \text{Percentage Increase} = \left( \frac{\Delta E}{E_i} \right) \times 100 \]
- Substitute the values:
\[ \text{Percentage Increase} = \left( \frac{29.2 \times 10^8}{81 \times 10^8} \right) \times 100 \]
- Simplifying gives:
\[ \text{Percentage Increase} \approx 36.04\% \]

### Conclusion:
When the temperature of the metal is raised from 27°C to 51.2°C, the rate at which it radiates energy increases by approximately **36.04%**.