The size of a certain insect population is given by $\mathrm{P}(\mathrm{t})=500 e^{02 \mathrm{t}}$, where $\mathrm{t}$ is measured in days. At what time will the population equal 1500 ? It will take $\square$ days for the population to equal 1500 . (Round to one decimal place as needed.)

Step 1 ：We are given the function \(P(t) = 500e^{0.2t}\) and we need to find the time \(t\) when the population \(P(t)\) equals 1500.

Step 2 ：This is a simple algebraic problem where we need to solve the equation \(500e^{0.2t} = 1500\) for \(t\).

Step 3 ：We can do this by first dividing both sides by 500 to get \(e^{0.2t} = 3\).

Step 4 ：Then, we can take the natural logarithm of both sides to get \(0.2t = \ln(3)\).

Step 5 ：Finally, we can solve for \(t\) by dividing both sides by 0.2.

Step 6 ：Doing this gives us \(t = 5.493061443340548\).

Step 7 ：Rounding to one decimal place, we find that it will take approximately \(\boxed{5.5}\) days for the population to equal 1500.