Problem 24 - Technical Mathematics with Calculus

Question

Population Growth: One of the most famous and controversial references to arithmetic and geometric progressions was made by Thomas Malthus in \(1798 .\) He wrote: "Population, when unchecked, increases in a geometrical ratio, and subsistence for man in an arithmetical ratio." Each day the size of a certain colony of bacteria is \(25 \%\) larger than on the preceding day. If the original size of the colony was 10,000 bacteria, find its size after 5 days.

Step-by-Step Solution

Answer

After 5 days, the size of the bacteria colony would be approximately 30,518 bacteria.

Step 1: Determine the growth factor

To find the growth factor for the bacteria population, convert the daily growth percentage to a decimal and add it to 1. Since the population grows by 25% each day, the growth factor is 1 + 0.25 = 1.25.

Step 2: Use the exponential growth formula

To calculate the size of the bacteria colony after a certain number of days, use the exponential growth formula: final size = initial size * (growth factor)^(number of days). In this case, the formula becomes: final size = 10,000 * (1.25)^5.

Step 3: Calculate the size after 5 days

Calculate the final size of the bacteria colony by using the formula from Step 2: final size = 10,000 * (1.25)^5 = 10,000 * 3.05175 = 30,517.5 bacteria. Since you cannot have a fraction of a bacterium, round this number to the nearest whole number.

Key Concepts

Arithmetic and Geometric Progressions

Starting with arithmetic and geometric progressions, it's essential to understand the distinction between these two fundamental concepts in mathematics. Arithmetic progression is a sequence of numbers in which each term is derived by adding a constant to the previous term. For instance, the sequence 2, 4, 6, 8, and so on, is an arithmetic progression where the constant added is 2.

In contrast, a geometric progression is a sequence where each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For example, the sequence 3, 6, 12, 24, ... is a geometric progression with a common ratio of 2.

The distinction becomes particularly important when considering growth patterns. Thomas Malthus' observation about population growth reflects a geometric progression, as populations tend to grow by proportion rather than by a constant addend, which is a characteristic of an arithmetic progression.

Exponential Growth Formula

Delving deeper into the mechanics of geometric progressions reveals the concept of exponential growth. The exponential growth formula encapsulates this idea and can be expressed as: \[ X_t = X_0 \times (1 + r)^t \]
where \(X_t\) is the final value after time \(t\), \(X_0\) is the initial value, \(r\) is the growth rate (expressed as a decimal), and \(t\) is the number of time intervals.

This formula is pivotal for a range of applications, from banking and finance to environmental science and demographics, because it helps to predict future values based on current data and growth rate. For the bacteria colony in the exercise, the exponential growth formula is instrumental to determine its size over time reliably.

Population Growth

The concept of 'population growth' typically refers to the change in the number of individuals in a population. When unchecked, as Malthus suggested, population growth follows a geometric progression, exhibiting patterns of exponential growth. In real-world scenarios, factors like food supply, disease, and habitat limitations can impose checks on this growth.

In the case of the bacteria colony from the exercise, the population growth is ideal, free from external constraints, allowing the exercise to illustrate the simplicity and predictability of exponential growth within a closed system. Through exercises such as these, students can better understand the implications of exponential growth in more complex ecosystems and human populations.

Daily Growth Percentage

Understanding 'daily growth percentage' involves interpreting a percentage as a decimal to calculate the daily increase in a population or value. In the case of the bacteria colony, a daily growth of 25% is critical. It's essential to add 1 to the converted decimal form of the growth rate to find the growth factor, which in this example is 1.25.

When applied to the exponential growth formula, this factor answers how much larger the population is each day compared to the previous day. This type of calculation is not only applicable to population dynamics but also to financial growth calculations, like interest compounding daily in a bank account. Hence, understanding daily growth percentage is crucial for projecting growth trajectories in various fields.