Step 1 :The size of a certain insect population is given by \(P(t)=100 e^{.03 t}\), where \(t\) is measured in days. We need to find out at what time will the population equal 300.
Step 2 :We need to solve the equation \(100 e^{.03 t} = 300\) for \(t\). This is an exponential equation, and we can solve it by taking the natural logarithm on both sides of the equation.
Step 3 :By solving the equation, we find that \(t\) is approximately 36.63.
Step 4 :Final Answer: The time at which the population will equal 300 is approximately \(\boxed{36.63}\) days.