Application of exponential and logarithmic functions Exponential growth and decay

Bacteria

Exponential growth In the laboratory, under favorable conditions, a growing bacterial population doubles at regular intervals. Growth is by geometric progression: 1, 2, 4, 8, etc. or 2 0, 2 1, 2 2, n (where n = the number of generations). This is called exponential growth. Starting with the formula for the nth term of a geometric sequence. We can define our key terms in the context of the problem. P 0 = U 1, the initial population P n = Un, the population after a certain time r = the reproductive rate of the bacterian = the number of generations This gives the following:

Creating a formula to determine the population at a certain time t. As our bacteria double in number over a set interval we have, r = 2 This gives the formula The number of generations is given as the time in hours divided by the generation time. The Mycobacterium tuberculosis bacterium doubles every hours Todar 2005). If we take the generation time as 20 then t = hours from initial reading.

Explaining how the formula works: If the initial population is 10 bacterium, then we can calculate how many bacteria will be present after 1 week. In total there are 168 hours in a week. This means the bacteria will reproduce 8.4 times We can also calculate how long it would take to reach a certain population size. When will the population reach 1 million? It would need days.

Real life … In reality, exponential growth is only part of the bacterial life cycle, and not representative of the normal pattern of growth of bacteria in Nature. Explain why this is.

Explore ….. Take a scenario that will lead to exponential growth and develop a formula that can model the situation using the geometric sequences. Write up your investigation in sections: (1) Introduction – necessary background to your scenario including why you have chosen this particular case. (2) Development of a formula. (3) Using the formula for find something out. (4) Small conclusion of your finding and any possible limitations.

References (1) (2) (3) (4) athogen%20Life%20Cycle.html athogen%20Life%20Cycle.html