Question

A one-liter jar would hold about \(10^{16}\) bacteria. Based on the number of doubling times in our 24 -hour experiment, show by calculation that our setup was woefully unrealistic: that even if we started with a single bacterium, we would have far more than \(10^{16}\) bacteria after 24 hours if doubling every 10 minutes.

Short answer

Question: In a hypothetical scenario, a bacterium doubles every 10 minutes. Explain why starting with a single bacterium, after 24 hours, there will be more bacteria than the capacity of a one-liter jar, which is approximately \(10^{16}\) bacteria. Answer: After calculating the total number of bacteria after 24 hours, we find that the total number of bacteria would be \(2^{144}\), which is approximately \(2.2 \times 10^{43}\) bacteria. This number is significantly greater than the capacity of a one-liter jar, which is approximately \(10^{16}\) bacteria. Therefore, even starting with a single bacterium, there would be far more than \(10^{16}\) bacteria after 24 hours, making our setup woefully unrealistic.

Step-by-step solution

Determine the number of minutes in 24 hours

To find out how many minutes are in 24 hours, we will multiply the number of minutes in an hour (60) by the number of hours (24). 24 hours × 60 minutes/hour = 1440 minutes

Calculate the number of doubling times in 24 hours

Since the bacteria double every 10 minutes, we will divide the total minutes in 24 hours (1440 minutes) by the doubling time (10 minutes/doubling). 1440 minutes ÷ 10 minutes/doubling = 144 doublings

Calculate the total number of bacteria after 24 hours

Given that we start with one bacterium, after 24 hours with 144 doublings, we need to calculate 2 raised to the power of 144. Total bacteria = \(2^{144}\)

Compare the total bacteria with the one-liter jar capacity

Now, we will compare the total number of bacteria after 24 hours (\(2^{144}\)) to the one-liter jar capacity, which is approximately \(10^{16}\) bacteria. \(2^{144} \approx 2.2 \times 10^{43} \gt 10^{16}\)

Conclusion

Since the total number of bacteria after 24 hours (\(2.2 \times 10^{43}\)) is much greater than the capacity of a one-liter jar (\(10^{16}\)), our setup is indeed woefully unrealistic. Even starting with a single bacterium, there would be far more than \(10^{16}\) bacteria after 24 hours if doubling every 10 minutes.

Key concepts

Bacterial Growth

Bacterial growth is a fascinating process where a bacterial population increases in number. It follows a characteristic pattern, often beginning exponentially. Starting with a single bacterium, these microorganisms replicate and double their population at consistent intervals, known as doubling times. This means that if a bacterium can double every 10 minutes, in just a handful of hours, the numbers skyrocket to levels that are hard to believe could fit into such a short time span.
Understanding bacterial growth is crucial for many fields, including medicine, environmental science, and food safety, as it provides insight into how bacterial colonies can expand rapidly under optimal conditions.
In our scenario, the bacteria double every 10 minutes, which is a huge growth rate. It's important because it shows how quickly they can multiply, leading to a massive number within a single day.

Mathematical Doubling

Mathematical doubling is a process where a quantity repeatedly multiplies by two, creating a rapid increase over time. In the case of bacterial growth, knowing the doubling time helps us predict the total population size after a certain period. This is expressed in power of twos, such as in our exercise where we use the formula:

• Total bacteria = \(2^{n}\)

Here, \(n\) is the number of times the bacteria have doubled.
For our exercise, after calculating that bacteria double 144 times in 24 hours, we find \(2^{144}\), which represents the massive scale of growth from a single bacterium to a huge population.
Understanding this concept helps students grasp the impact of exponential growth in real-world situations, showcasing how quickly numbers can swell when doubling constantly.

Scientific Experimentation

Scientific experimentation is a systematic way to test hypotheses and observe phenomena under controlled conditions. It provides a structured means of understanding complex concepts, such as the bacterial growth in our exercise.
In the experiment outlined, we start with an initial hypothesis: a certain environment allows bacteria to grow at a consistent rate, doubling every 10 minutes. By calculating the total possible growth over 24 hours, the experiment demonstrates how theoretical predictions can vastly exceed practical limitations, illustrating why a one-liter jar capacity was underestimated.
This kind of experimentation is foundational to scientific research because it involves:

• Forming a hypothesis or question
• Conducting experiments or simulations
• Collecting and analyzing data
• Drawing conclusions

Through such structured approaches, scientists can better comprehend the rules governing phenomena like microbial proliferation, enabling advancements in fields like microbiology and beyond.