Final answer:
The population growth model for the given bacteria population is P(t) = 674 * e^(0.047t). The population exactly 3 weeks from now can be calculated by plugging in t = 21 into the growth model. The rate of change in the population at that time can be found by differentiating the growth model and evaluating it at t = 21. The time it takes for the population to reach 6022 can be found by setting the population equation equal to 6022 and solving for t.
Step-by-step explanation:
The population growth model for the given population of bacteria is given by the equation P(t) = P0 * e^(kt). Here, P(t) represents the population after t days, P0 is the initial population, k is the growth rate, and e is the base of the natural logarithm.
To find the growth model, we need to determine the values of P0 and k. We are given that the current population is 674. Plugging in this value into the equation, we get 674 = P0 * e^(k*0). This simplifies to P0 = 674.
We are also given that the relative (daily) growth rate is 4.7%. This means that k = 0.047. Therefore, the growth model is P(t) = 674 * e^(0.047t).
To find the population exactly 3 weeks from now, we substitute t = 21 (3 weeks = 21 days) into the growth model and calculate the population using a scientific calculator or a computer software. Round the result to the nearest bacterium to get the approximate population.
To find the rate of change in the population exactly 3 weeks from now, we differentiate the growth model with respect to t and evaluate it at t = 21. This will give us the instantaneous rate of change in the population at that time point.
To find when the population will reach 6022, we set the population equation equal to 6022 and solve for t. This will give us the number of days it takes for the population to reach 6022 starting from the initial population.