Question

If a one-liter jar holds \(10^{16}\) bacteria, how many bacteria would we start in the jar so that the jar reaches full capacity after 24 hours if we increase the doubling time to a more modest/realistic 30 minutes?

Short answer

Answer: The initial number of bacteria should be \(10^{16}\) / 2^((24 * 60) / 30).

Step-by-step solution

Calculate the number of doubling intervals

As the doubling time is 30 minutes and we have 24 hours before the jar reaches its full capacity, we first need to calculate the total number of doubling intervals in the given timeframe. Since there are 60 minutes in an hour and we have 24 hours, there are (24 * 60) minutes in total. Divide this by the doubling time to find the number of intervals: Number of doubling intervals = (24 * 60) / 30

Calculate the initial number of bacteria

Now, we will use the exponential growth formula to find the initial number of bacteria. The exponential growth formula is: Final amount = Initial amount * 2^(number of doubling intervals) We are given the final amount as \(10^{16}\) bacteria, and we need to find the initial amount. Rearrange the formula to solve for the initial amount: Initial amount = Final amount / 2^(number of doubling intervals) Substitute the given values into the formula and calculate the initial amount of bacteria: Initial amount = \(10^{16}\) / 2^((24 * 60) / 30) The initial number of bacteria that should be present in the jar to reach full capacity after 24 hours with a doubling time of 30 minutes is the calculated value of the initial amount.

Key concepts

Doubling Time

Doubling time refers to the period it takes for a quantity to double in size or value at a constant growth rate. In the context of bacteria growth, this is the time it takes for the bacteria population to double. This is a crucial concept when studying exponential growth, as it helps in predicting how quickly the population will increase. To determine doubling time in real-life scenarios such as this exercise, one should identify:

• The current population size.
• The expected future population size.
• The time it takes to achieve that population growth.

With these often predictable parameters, using the formula for exponential growth allows for determining the number of intervals where doubling occurs. By dividing the total available time by the given doubling time (30 minutes in our case), one can calculate the total number of times the population will double in the specified period.

Bacteria Population

Bacteria are microorganisms that can rapidly reproduce through asexual reproduction, commonly by cell division known as binary fission. This process perfectly illustrates exponential growth, where the population continuously doubles at regular intervals.As seen in the exercise, understanding the bacteria population's growth over time is critical. By identifying:

• The initial number of bacteria (the starting amount)
• The desired final goal for the population size (in this case, the jar capacity of \(10^{16}\) bacteria)
• The doubling time

We can estimate how much the bacteria population will grow over any given period. This step is essential for making predictions in biological studies and microbiology-related industries, where managing microbial growth is vital.

Mathematical Modeling

Mathematical modeling is the process of using mathematical structures and methods to represent real-world situations. In our case, it helps describe the expanding bacteria population over time. Such models are crucial because:

• They provide a simplified way to predict complex systems like bacterial growth.
• They help visualize the growth pattern using formulas and equations, facilitating better understanding.

For bacteria population growth, the exponential growth formula is a standard model. It is expressed as:\[\text{Final amount} = \text{Initial amount} \times 2^{\text{(number of doubling intervals)}}\]To find an initial amount needed to reach a desired final population, the formula can be rearranged. Here's how the calculation unfolds:First, find out how many doubling intervals fit into the total period:\[(24\, \text{hours} \times 60\, \text{minutes/hour}) / 30\, \text{minutes/doubling interval}\]Then, utilize the rearranged exponential model to find the initial amount:\[\text{Initial amount} = \frac{10^{16}}{2^{\text{(number of doubling intervals)}}}\]Incorporating these formulas allows individuals to preemptively manage biological growth scenarios strategically.