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Chapter 4: Problem 16

Growth of bacteria: The organism \(E\). coli is a common bacterium. Under certain conditions it undergoes cell division approximately each \(20 \mathrm{~min}-\) utes. During cell division, each cell divides into two cells. a. Explain why the number of \(E\). coli cells present is an exponential function of time. b. What is the hourly growth factor for E. coli? c. Express the population N of E. coli as an exponential function of time t measured in hours. (Use N0 to denote the initial population.) d. How long will it take a population of E. coli to triple in size?

Short Answer

Expert verified
E. coli population is exponential; triples in approx. 0.528 hours.

Step by step solution

01

Understanding Exponential Growth

Exponential growth occurs when the growth rate of a population or process is proportional to its current value. For the E. coli bacterium, each cell divides into two cells every 20 minutes. This doubling process every fixed period (20 minutes) is characteristic of exponential growth.
02

Calculating Hourly Growth Factor

E. coli cells double every 20 minutes. In one hour, there are three 20-minute periods. Thus, the hourly growth factor is calculated as: \[ 2^3 = 8 \]. This means the population grows by a factor of 8 every hour.
03

Expressing the Population as an Exponential Function

Let \( N(t) \) be the population at time \( t \) hours and \( N_0 \) be the initial population. Since the population grows by a factor of 8 each hour, the function can be expressed as: \[ N(t) = N_0 imes 8^t \].
04

Finding Time to Triple the Population

To determine how long it takes for the population to triple, set \( N(t) = 3N_0 \). Using the function \( N(t) = N_0 imes 8^t \), set \( 3N_0 = N_0 imes 8^t \). Simplifying, \( 3 = 8^t \). Solve for \( t \) by taking logarithms: \[ t = \frac{\log{3}}{\log{8}} \approx 0.528 \text{ hours}. \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Bacteria Growth
Bacteria growth is an example of exponential growth, a concept where the rate of increase of a population is proportional to the population itself. This means, the larger the population, the faster it grows. With bacteria like _E. coli_, this is evident as each bacterium divides into two, replicating completely every 20 minutes under optimal conditions.

To visualize this, imagine you start with a single _E. coli_ bacterium. After 20 minutes, it divides, and you have two. Following another 20 minutes, the two divide to make four, and the doubling process continues. This doubling pattern is what creates the exponential curve seen in their population growth over time.

In this growth process:
  • The bacteria double its number in a fixed time frame (20 minutes here).
  • Each new generation is larger than the previous, leading to rapid population growth.
Cell Division
Cell division in bacteria like _E. coli_ is a simple process known as binary fission. During binary fission, a single bacterium grows to a certain size and then divides into two equal-sized daughter cells. This happens in a continuous cycle, enabling rapid population expansion.

Here are some key points about bacterial cell division:
  • Each cell divides into two, leading to a doubling of the population.
  • The division happens every 20 minutes for _E. coli_ under favorable conditions.
  • No genetic variation occurs during this process; the daughter cells are clones of the parent cell.
Understanding this mechanism is crucial, as it drives the exponential growth pattern of bacterial cultures.
Hourly Growth Factor
The hourly growth factor represents how much a population increases in one hour. In our example with _E. coli_, every 20 minutes the population doubles. As an hour contains three 20-minute intervals, the population doubles three times in one hour.

Calculating the hourly growth factor involves determining the total increase over these three intervals:
  • First 20 minutes: doubles once, so growth multiplies by 2.
  • Second 20 minutes: doubles again, resulting in \( 2 \times 2 = 4 \).
  • Third 20 minutes: doubles a third time \( 4 \times 2 = 8 \).
Thus, the hourly growth factor of _E. coli_ is \( 2^3 = 8 \), which means the population increases by a factor of 8 every hour under ideal growth conditions.
Exponential Function
An exponential function is a mathematical representation of exponential growth. For _E. coli_, the population at any given time can be calculated using such a function. This allows us to precisely predict future population sizes if we know the current size and growth rate.

The exponential function for the population of _E. coli_ over time is expressed as:\[ N(t) = N_0 \times 8^t \]where:
  • \( N(t) \) is the population size at time \( t \) hours.
  • \( N_0 \) is the initial size of the population.
  • \( 8 \), the hourly growth factor, means the population increases 8 times every hour.
This function is powerful because it can be rearranged to solve for different variables, such as determining how long it takes for the population to triple in size or identifying the population at a future time.

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Most popular questions from this chapter

Wages: A worker is reviewing his pay increases over the past several years. The table below shows the hourly wage W, in dollars, that he earned as a function of time t, measured in years since the beginning of 1990. $$ \begin{array}{|c|c|} \hline \text { Time } t & \text { Wage } W \\ \hline 1 & 15.30 \\ \hline 2 & 15.60 \\ \hline 3 & 15.90 \\ \hline 4 & 16.25 \\ \hline \end{array} $$ a. By calculating ratios, show that the data in this table are exponential. (Round the quotients to two decimal places.) b. What is the yearly growth factor for the data? c. The worker can't remember what hourly wage he earned at the beginning of 1990 . Assuming that \(W\) is indeed an exponential function, determine what that hourly wage was. d. Find a formula giving an exponential model for \(W\) as a function of \(t\). e. What percentage raise did the worker receive each year?f. Given that prices increased by 34% over the decade of the 1990s, use your model to determine whether the worker鈥檚 wage increases kept pace with inflation.

Nearly linear or exponential data: One of the two tables below shows data that are better approximated with a linear function, and the other shows data that are better approximated with an exponential function. Make plots to identify which is which, and then use the appropriate regression to find models for both. $$ \begin{aligned} &\begin{array}{|c|c|} \hline t & f(t) \\ \hline 1 & 3.62 \\ \hline 2 & 23.01 \\ \hline 3 & 44.26 \\ \hline 4 & 62.17 \\ \hline 5 & 83.25 \\ \hline \end{array}\\\ &\begin{array}{|c|c|} \hline t & g(t) \\ \hline 1 & 3.62 \\ \hline 2 & 5.63 \\ \hline 3 & 8.83 \\ \hline 4 & 13.62 \\ \hline 5 & 21.22 \\ \hline \end{array} \end{aligned} $$

Cell phones: The following table shows the number, in millions, of cell phone subscribers in the United States at the end of the given year. $$ \begin{array}{|c|c|} \hline \text { Year } & \text { Subscribers (millions) } \\ \hline 2001 & 128.4 \\ \hline 2002 & 140.8 \\ \hline 2003 & 158.7 \\ \hline 2004 & 182.1 \\ \hline 2005 & 207.9 \\ \hline \end{array} $$ a. Plot the data points. Does this plot make it look reasonable to approximate the data with an exponential function? b. Use exponential regression to construct an exponential model for the subscriber data. c. Add the graph of the exponential model to the plot in part a. d. What was the yearly percentage growth rate from the end of 2001 through the end of 2005 for cell phone subscribership? e. In 2005 an executive had a plan that could make money for the company, provided that there would be at least 250 million cell phone subscribers by the end of 2007. Solely on the basis of an exponential model for the data in the table, would it be reasonable for the executive to implement the plan?

Aphid growth on broccoli: This problem and the following one are based on the results of a study by R. Root and A. Olson of population growth for aphids on various host plants.44 The value of r = 0.243 per day is valid for aphids reared on broccoli plants outdoors. The table on the next page describes the growth of an aphid population reared on broccoli plants indoors (in a controlled environment). Time t is measured in days. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { Time } t & 0 & 1 & 2 & 3 & 4 \\ \hline \text { Population } & 25 & 30 & 37 & 44 & 54 \\ \hline \end{array} $$ Calculate r for this table. Did the aphids fare better indoors or outdoors?

Research project: For this project, you should collect and analyze data for a population of M\&M's TM. Start with four candies, toss them on a plate, and add one for each candy that has the M side up; record the data. Repeat this seven times and see how close the data are to being exponential. For a detailed description, go to http://college.hmco.com/PIC/crauder4e.
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