1. A population of rats follow the logistic model of population growth. The carrying capacity of the population is 10,000. The initial population is 1,500 and after 5 weeks, it is 4,000.
a. Write the logistic differential equation for the given data.
b. Find the formula for the weekly growth of population. What is the population in 10 weeks? When will the population be 9,000?
2. The rate at which an epidemic spreads through a community with 2000 susceptible residents is jointly proportional to the number of residents who have been infected and the number of susceptible residents who have not yet been infected by the disease. Suppose that 500 residents had the disease initially and 855 residents had been infected by the end of the first week. After how many weeks will 1000 residents be infected?
3. A freshly brewed cup of coffee has temperature 95 degrees Celsius in a 20 degrees Celsius room. When its temperature is 70 degrees Celsius, it is cooling at the rate of 1 degree Celsius per minute. What is the temperature of the coffee after t seconds?
4. A population grows weekly according to the logistic law. At the start, there are 200 individuals in the population. Write the differential equation describing the rate of change of population per week if the carrying capacity of the population is 3000 individuals and k=0.06. Find the population after 30 weeks.
5. One model for the spread of an epidemic is that the rate of spread is jointly proportional to the number of infected people and the number of uninfected people. In an isolated town of 5,000 inhabitants. 160 people have a disease at the beginning of the week and 1,200 have it at the end of the week. How long does it take for 80% of the population to become infected?
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