Problem 7 - Thomas Calculus

Question

Cholera bacteria Suppose that the bacteria in a colony can grow unchecked, by the law of exponential change. The colony starts with 1 bacterium and doubles every half-hour. How many bacteria will the colony contain at the end of 24 hours? (Under favorable laboratory conditions, the number of cholera bacteria can double every 30 \(\mathrm{min.}\) In an infected person, many bacteria are destroyed, but this example helps explain why a person who feels well in the morning may be dangerously ill by evening.)

Step-by-Step Solution

Answer

281,474,976,710,656 bacteria after 24 hours.

Step 1: Identify the doubling period

The bacteria double every 30 minutes, which means that each doubling period is half an hour.

Step 2: Calculate the total number of doubling periods in 24 hours

There are 24 hours in a day, and since each hour contains 2 half-hours, there are a total of \(24 \times 2 = 48\) doubling periods in 24 hours.

Step 3: Use the exponential growth formula

The exponential growth formula is \( N = N_0 \times 2^n \), where \( N_0 \) is the initial number of bacteria and \( n \) is the number of doubling periods.

Step 4: Plug the values into the formula

Here, \( N_0 = 1 \), and \( n = 48 \). Plug these values into the formula: \[ N = 1 \times 2^{48} \].

Step 5: Compute the final number of bacteria

Calculate \( 2^{48} \) to find that \[ N = 281,474,976,710,656 \]. Thus, the colony will contain approximately 281 trillion bacteria at the end of 24 hours.

Key Concepts

Doubling Period

The doubling period is a simple yet powerful concept in understanding exponential growth. It refers to the time it takes for a quantity to double in size or amount. In the context of bacterial growth, the doubling period is the time frame in which a population of bacteria doubles.
For the cholera bacteria example, the doubling period is every 30 minutes. This means, if you start with one bacterium, you will have two after 30 minutes, four after another 30 minutes, and so on.
Recognizing doubling periods is essential for predicting how quickly a population can grow.

• It helps us understand the speed and potential impact of growth.
• It allows for calculating the future size of populations based on periodic increases.

This principle of doubling can be applied to many fields beyond biology, such as finance, where investments compound, or technology, where processing power doubles.

Bacterial Growth

Bacterial growth is a rapid and dramatic illustration of exponential growth, especially under favorable conditions where resources are abundant and checks like predation or competition are absent.
In the scenario provided, we see how starting with just a single bacterium can lead to an explosive increase in a relatively short time. This type of growth is characterized by:

• A constant relative growth rate, meaning the population increases by a specific factor (in this case, doubles) over equal time intervals.
• Unchecked reproduction, resulting in a large population quickly if conditions remain ideal.

The concept aids in understanding public health concerns, especially how bacterial infections can seem to arise suddenly. It highlights the importance of early interventions and preventive measures to control bacterial outbreaks before they become widespread.

Exponential Change

Exponential change refers to a process of change where the rate of growth or decay is proportional to the current amount. In essence, as the quantity grows, the rate at which it grows becomes faster, leading to dramatic increases over time.
This is precisely what is demonstrated in the bacterial growth example. As the number of bacteria increases, the amount added each doubling period also exponentially increases, compounding the total rapidly.
Understanding exponential change is crucial because:

• It helps in modeling real-world phenomena like population growth, radioactive decay, and interest calculations in finance.
• It brings insight into the challenges of predicting outcomes when growth occurs at a non-linear rate.

Encountering exponential growth in practical scenarios requires understanding how quickly changes can happen and preparing for their potential impacts.