Problem 48 - College Algebra and Calculus: An Applied Approach

Question

Suppose that you have a single imaginary bacterium able to divide to form two new cells every 30 seconds. Make a table of values for the number of individuals in the population over 30 -second intervals up to 5 minutes. Graph the points and use a graphing utility to fit an exponential model to the data.

Step-by-Step Solution

Answer

The bacterium population growth can be represented as an exponential growth model. Over five minutes the population grows rapidly during each 30-second interval.

Step 1: Understand and Note Down Information

The starting point is a single bacterium which splits into two every 30 seconds. Therefore after 30 seconds, we have 2 bacteria. The time being monitored is up to 5 minutes, which is equal to 10 intervals of 30 seconds.

Step 2: Create the Table of Values

Using the given information, the number of bacteria for each 30-second interval can be calculated by doubling the previous interval's value. Begin with 0 interval (0-second) with 1 bacterium, and subsequently, for each 30-second interval double the number from the previous interval until 5 minutes (10 intervals).

Step 3: Represent the Values on a Graph

Set up a graph with 'Time (in 30-second intervals)' on the x-axis and 'Number of Bacteria' on the y-axis. Plot the points according to the table values from step 2. Because it is stated that this bacterium's growth rate is exponential, the graph should start to curve upwards as time progresses.

Step 4: Use a Graphing Utility to Fit an Exponential Model

Utilize a graphing utility (like Desmos or GeoGebra) to determine the best fit exponential model for the data. The points from the graph should fall along this curve indicating a good model fit.

Step 5: Interpret the Graph

The graph of the exponential growth model indicates that the population of the bacterium grows rapidly as time progresses. Each point on the graph represents the population of the bacteria at the given time, showing exponential growth.