The temperature of a radiation body increases by `30 %`. Then the increase in the amount of radiation is?

Options

A. `185 %`
B. `285%`
C. `325%`
D. `130 %`

Explanation

To solve the problem of how much the amount of radiation increases when the temperature of a radiating body increases by 30%, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between radiation and temperature**: According to Stefan-Boltzmann law, the amount of radiation emitted by a body is directly proportional to the fourth power of its absolute temperature. This can be expressed as:
\[ E \propto T^4 \]
where \(E\) is the amount of radiation and \(T\) is the temperature.

2. **Define the initial and final temperatures**: Let the initial temperature be \(T\). If the temperature increases by 30%, the new temperature \(T'\) can be expressed as:
\[ T' = T + 0.3T = 1.3T \]

3. **Calculate the initial and final amounts of radiation**: The initial amount of radiation \(E\) at temperature \(T\) is:
\[ E = kT^4 \]
where \(k\) is a constant of proportionality. The new amount of radiation \(E'\) at temperature \(T'\) is:
\[ E' = k(T')^4 = k(1.3T)^4 \]

4. **Expand the expression for \(E'\)**:
\[ E' = k(1.3^4)(T^4) = 1.3^4 \cdot kT^4 = 1.3^4 \cdot E \]

5. **Calculate \(1.3^4\)**:
\[ 1.3^4 = 2.8561 \quad (\text{approximately}) \]
Therefore, we can write:
\[ E' \approx 2.8561E \]

6. **Determine the increase in radiation**: The increase in radiation \(\Delta E\) can be calculated as:
\[ \Delta E = E' - E = (2.8561E - E) = 1.8561E \]

7. **Calculate the percentage increase in radiation**: The percentage increase in radiation can be calculated using the formula:
\[ \text{Percentage Increase} = \left(\frac{\Delta E}{E}\right) \times 100 = \left(\frac{1.8561E}{E}\right) \times 100 \]
Simplifying this gives:
\[ \text{Percentage Increase} \approx 185.61\% \]

### Final Answer:
The increase in the amount of radiation is approximately **185.61%**.