Step 1 :Given the intensity of light, \(I\), in watts per meter-squared, is inversely proportional to the distance squared and given by the equation \(I=\frac{13}{x^{2}}\), where \(x\) is meters from the light source.
Step 2 :We are asked to find the distance from the light source where the intensity is at least 18.77 watts per meter-squared. This means we need to solve the equation \(I=\frac{13}{x^{2}}\) for \(x\) when \(I\) is 18.77.
Step 3 :We can rearrange the equation to \(x=\sqrt{\frac{13}{I}}\) and substitute \(I\) with 18.77 to find \(x\).
Step 4 :Substituting \(I\) with 18.77, we get \(x = \sqrt{\frac{13}{18.77}}\) which simplifies to \(x = 0.8322226659953956\).
Step 5 :Rounding to two decimal places, we get \(x = 0.83\).
Step 6 :Final Answer: The intensity is at least 18.77 watts per meter-squared at a distance of approximately \(\boxed{0.83}\) meters from the light source.