Bacterial Growth. Suppose that a biologist performs an...

Bacterial Growth. Suppose that a biologist performs an experiment in which he or she measures the rate at which a specific type of bacterium reproduces asexually in different culture media. The experiment shows that in Medium A the bacteria reproduce once every 60 minutes, and in Medium B the bacteria reproduce once every 90 minutes. Assume that a single bacterium is placed on each culture medium at the beginning of the experiment. Write a program that calculates and plots the number of bacteria present in cach culture at intervals of 3 hours from the beginning of the experiment antil 24 hours have elapsed. Make two plots, one a linear xy plot and the other a linear-log (semi logy) plot. How do the numbers of bacteria compare on the two media after 24 hours?

Key Concepts

Semi-Log Plotting

Semi-log plotting uses a logarithmic scale for one axis (typically the y-axis), which is particularly useful when depicting exponential growth. This transformation converts multiplicative relationships into additive ones, rendering the exponential growth as a straight line, thereby making it easier to compare different growth rates and detect proportional differences across orders of magnitude.

Linear Plotting

Linear plotting involves graphing data using a standard xy-coordinate system where both axes are scaled linearly. This type of plot represents the absolute changes in bacterial population over time, but can become visually compressed when dealing with exponential increases, potentially masking important trends.

Discrete-Time Sampling

Discrete-time sampling involves taking measurements at specific, regular time intervals rather than continuously monitoring the process. In experiments, recording bacterial populations every 3 hours allows for a simplified analysis of growth patterns over a defined period (such as 24 hours), translating a continuous exponential process into a series of data points.

Exponential Growth

Exponential growth describes processes where the rate of increase is proportional to the current value. In the context of bacterial reproduction, this means that the number of bacteria increases multiplicatively over time, making the growth rapid as each bacterium divides independently. This concept forms the foundation for modeling populations that double or otherwise multiply at regular intervals.

Doubling Time

Doubling time is the period required for a quantity undergoing exponential growth to double in size. In biological contexts, it is defined by the reproduction rate of the organism. A shorter doubling time, such as one bacterium reproducing every 60 minutes, indicates a faster growth rate compared to a longer doubling time, such as reproduction every 90 minutes.

Transcript

00:01 All right, in this problem, we're going to use a graphing calculator and take the data about the bacteria count over time and type it into our statistics lists.

00:10 So we press the stat button, and then we go into edit, and then we type the numbers into list one and list two.

00:16 And the first thing we want to do with it is make a scatter plot.

00:19 So we need to go into the stat plot window, which is second y -equals, go into the menu for stat plot one and turn it on, and make sure it's a scatter -plot using list one and list.

00:31 List two.

00:32 The next thing we can do is set a good window for our plot so we can go to zoom and then down to zoom 9, which is zoom stat.

00:41 So there's our scatter plot.

00:43 We can see it's curving like an exponential growth function would.

00:47 And the next thing we want to do is find the exponential model.

00:51 So we can go back to stat, over to calculate, and go down until we find exponential regression, select that, use list one and list two, and then let's store the regression equation in our y equals menu.

01:06 So from here we go to the vars button, over to y variables, choose function, and choose y1.

01:12 And now we can calculate.

01:14 So these are the values that would go into our exponential model.

01:18 We have approximately y equals a, or y equals 36 .89 times 1 .07 to the x power...

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