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Chapter 24: Problem 47

The doubling time for \(E .\) coli is 20 minutes, and you start getting sick when just 10 bacteria enter your system. How many bacteria are in your body after 12 hours?

Short Answer

Expert verified
687,194,767,360 bacteria

Step by step solution

01

- Understand the Doubling Formula

To determine the number of bacteria after a certain period, we use the doubling formula: \[ N(t) = N_0 \times 2^{\frac{t}{d}} \]where \(N(t)\) is the number of bacteria at time \(t\) minutes, \(N_0\) is the initial number of bacteria, and \(d\) is the doubling time.
02

- Identify the Variables

Given: \(N_0 = 10\) (initial bacteria)\(d = 20 \text{ minutes}\) (doubling time)\(t = 12 \text{ hours} = 720 \text{ minutes}\) (time period).
03

- Substitute Values into the Formula

Plug the given values into the formula:\[ N(720) = 10 \times 2^{\frac{720}{20}} \]
04

- Simplify the Exponent

Simplify the exponent:\[ \frac{720}{20} = 36 \]Therefore, the formula becomes:\[ N(720) = 10 \times 2^{36} \]
05

- Calculate the Number of Bacteria

Calculate \(2^{36}\):\[ 2^{36} = 68,719,476,736 \]Multiply this by 10:\[ N(720) = 10 \times 68,719,476,736 = 687,194,767,360 \]

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Key Concepts

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

Doubling Time
Doubling time is the period it takes for a quantity to double in size or value. In bacterial growth, it refers to the time needed for a population of bacteria to double. For example, the doubling time for E. coli is 20 minutes.
This means if you start with 10 bacteria, after 20 minutes, there will be 20 bacteria, after another 20 minutes, there will be 40 bacteria, and so on.
The formula to calculate the number of bacteria, given the doubling time, is:
The key variables in this formula are:

  • Plugging these into our formula helps understand how fast and how many bacteria grow over specific periods.
Exponential Growth
Exponential growth is a process where the quantity increases over time at a rate proportional to its current value. In the case of bacterial growth, the population of bacteria doubles after every fixed interval (doubling time). This results in a rapid increase in the number of bacteria.
For example, starting with 10 E. coli bacteria and a doubling time of 20 minutes, after 12 hours (or 720 minutes), we would have around 687,194,767,360 bacteria.
This rapid increase is due to the bacteria's ability to reproduce quickly, causing an exponential or logarithmic rise in their numbers.
The calculation involves understanding and using the doubling formula:
Using this formula, we can observe how quickly an initially small number of bacteria can grow into an enormous population.
This represents the power of exponential growth clearly.
E. coli
E. coli (Escherichia coli) is a type of bacteria commonly found in the intestines of humans and animals. Most strains are harmless, but some can cause severe foodborne illnesses. Understanding the growth rate of E. coli is vital in fields ranging from healthcare to food safety.
These bacteria are known for their rapid growth rate. Under optimal conditions, E. coli can double every 20 minutes. This makes it a prime candidate for studying bacterial growth patterns and the effects of antibiotics.
Scientists often use E. coli as a model organism in genetic and microbiological research due to its well-understood genetics and fast growth.
Knowing how quickly E. coli can grow helps in managing infections and devising strategies for controlling bacterial growth in various settings.

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Most popular questions from this chapter

The study of life and the study of astronomy are connected because (select all that apply) a. life may be quite commonplace in the universe. b. studying other planets may help explain why there is life on Earth. c. explorations of extreme environments on Earth suggest where to look for life elsewhere. d. life is a structure that evolved through physical processes, and life on Earth may not be unique. e. life elsewhere is most likely to be found by astronomers.

Why do scientists generally talk about molecules such as DNA or RNA when discussing life on Earth?

T/F: Scientists know with certainty that in the Solar System, life exists only on Earth.

The Kepler mission is currently searching for planets in the habitable zones of stars. Explain which factors in the Drake equation are affected by this search, and how the final number \(N\) will be impacted if Kepler finds that most stars have planets in their habitable zones.

The difference between a prokaryote and a eukaryote is that prokaryotes a. have no DNA. b. have no cell wall. c. have no nucleus. d. do not exist today.
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