Step 1: Understand the inverse square law for isotropic sources. The intensity \(I\) of radiation from an isotropic source is inversely proportional to the square of the distance \(r\) from the source, expressed as \( I \propto \frac{1}{r^2} \). This implies that \(I \cdot r^2\) is a constant, so \(I_1 r_1^2 = I_2 r_2^2\).
Step 2: List the provided data.
- Initial intensity, \(I_1 = 0.250\) W/m\(^2\).
- Initial distance, \(r_1 = 15\) m.
- Final distance, \(r_2 = 75\) m.
- The objective is to determine the final intensity, \(I_2\).
Step 3: Calculate \(I_2\).
Using the formula \(I_2 = I_1 \left( \frac{r_1}{r_2} \right)^2\):
\[ I_2 = 0.250 \, \text{W/m}^2 \times \left( \frac{15 \, \text{m}}{75 \, \text{m}} \right)^2 \]
\[ I_2 = 0.250 \times \left( \frac{1}{5} \right)^2 = 0.250 \times \frac{1}{25} \]
\[ I_2 = \frac{0.250}{25} = 0.010 \, \text{W/m}^2 \]