The Problem

The problem: A type of bacteria that inhabits the intestines of animals is named E. coli (Eschenchia coli). These bacteria are capable of rapid growth and can be dangerous to humans -- especially children. In one study, E. coli bacteria were capable of doubling in number every 49.5 minutes. Their number N after x minutes could be modeled by Suppose = 500,000 in the initial number of bacteria per milliliter. (Source: G. S.Stent, Molecular Biology of Bacterial Viruses.)

b). Determe graphically the elapsed time when there were 25 million bacteria per milliliter.

Solution to part b:

b). In this part of the problem, N is given to be 25,000,000. Substituting into the equation, , gives:

To solve for x graphically using the Intersections of Graphs Method, first change the WINDOW settings to the following:
XMin = 0, XMax = 400, Xscl = 100, YMin = 0, YMax = 30000000, and Yscl = 10000000. Now press and enter the functions into and as follows: = 25000000 and = . Press GRAPH and the graph should appear as shown below.

To solve for x, find the point of intersection. Press 2nd, then TRACE, and select 5. When asked for First curve?, using the ARROW KEYS, move the cursor so that it is near the point of intersection and press ENTER. When asked for Second curve?, move the cursor so that it is near the point of intersection and press ENTER. When asked for the Guess, just press ENTER again.

From the last graph above, the point of intersection is approximately (279.43021, 25000000). Thus, the bacteria has increased to 25000000 per milliliter after approximately x = 279 minutes. To convert minutes to hours, divide the number of minutes by 60. So 279 minutes could be expressed as 4.65 hours. To convert 4.65 hours to hours and minutes, multiply 0.65 by 60 to get 39 minutes.

The answer could be given as x = 279 minutes, or x = 4.65 hours, or x = 4 hours 39 minutes.