Sequences -- Population Models -- Stretch Your Understanding

1. Suppose that the population growth for a particular city can be described by an exponential model and that the population was 4,000,000 in 1990 and 4,200,000 in 1994. What will the population be in 1998? in 2000? What was the population in 1991? in 1992? in 1993?

2. The graph below shows the height of a bouncing ball. Height is measured along the y-axis and time along the x-axis. Measure the height of the ball at the top of each bounce. Based on these measurements do you think an exponential model would do a good job of describing the height of each bounce? If so, predict the height of the ball for the next three bounces.

   

3. Do you think exponential models are reasonable models for population growth? Why? Or why not?

4. Some atoms are unstable and spontaneously emit radiation and change into different atoms. This process is called radioactive decay. Consider a particular kind or isotope of atom that is unstable. Suppose that the probability that any one particular atom of this kind decays in one day is 5%. Thus after one day 95% of the original atoms of this kind remain. If we let A(0) = T be the original number of these atoms and A(1) = 0.95 T be the number remaining after one day then do you think an exponential model would describe the sequence A(n) representing the number of atoms remaining after n days? Explain.

5. In the early stages of an epidemic the infected population often grows exponentially -- for example, if each infected person infects 1.2 people every day and no one ever recovers then the change in the infected population can be described by the change equation

   I(n + 1) = 2.2 I(n)

   Where did the 2.2 come from? Why isn't the change equation

   I(n + 1) = 1.2 I(n)?

   If the current number of infected people is 2,000 in how many days will the infected population reach 10,000?

   Do you think this model will continue to be useful after many days? Why? Or why not?