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

Answer

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Answer

Rounding to one decimal place, we find that it will take approximately $5.5$ days for the population to equal 1500.

Steps

Step 1 ：We are given the function $P (t) = 500 e^{0.2 t}$ 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 $500 e^{0.2 t} = 1500$ for $t$ .

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

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

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 $5.5$ days for the population to equal 1500.