During optimal conditions, the rate of change of the population of a certain organism is proportional to the population at time t, in hours. At time t = 0 hours, the population is 300. At time t = 24 hours, the population is 1000. At what time t is the population 500?

(A) t = 2112

(B) t = 24 In 500 In 1000

(D) t = 300e (9)

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Answer:The population is of 500 after 10.22 hours.

Step-by-step explanation:The rate of change of the population of a certain organism is proportional to the population at time t, in hours.This means that the population can be modeled by the following differential equation:

In which r is the growth rate.

Solving by separation of variables, then integrating both sides, we have that:

Applying the exponential to both sides:

In which K is the initial population.

At time t = 0 hours, the population is 300.This means that K = 300. So

At time t = 24 hours, the population is 1000.This means that P(24) = 1000. We use this to find the growth rate. So

So

At what time t is the population 500?This is t for which P(t) = 500. So

The population is of 500 after 10.22 hours.