Problem 15 - Calculus: Early Transcendental Functions

Question

Involve exponential growth. Suppose that a population of \(E .\) coli doubles every 20 minutes. A treatment of the infection removes \(90 \%\) of the \(E .\) coli present and is timed to accomplish the following. The population starts at size \(10^{8},\) grows for \(T\) minutes, the treatment is applied and the population returns to size \(10^{8} .\) Find the time \(T\)

Step-by-Step Solution

Answer

The time \(T\) it takes for the population to return back to its original size after going undergoing the treatment is approximately equal to \( 20 \cdot log_2 (10) \) minutes.

Step 1: Write Out the Exponential Growth Equation

The exponential growth equation we need to start with is \( N(t) = N_0 \cdot 2^{(t/T)} \) where \( N_0 \) is the initial population, \( t \) is the time, and \( T \) is the time it takes for the population to double. We know that \( N_0 = 10^{8} \), \( T = 20 \) minutes and \( t = T \) (which we are trying to find). So our growth equation is \( N(T) = 10^{8} \cdot 2^{(T/20)} \).

Step 2: Solve for Population at Time T

Before the treatment is applied, the population grows to \( N(T) = 10^{8} \cdot 2^{(T/20)} \). After the treatment, the population reduces by \( 90\% \) and hence the population becomes \( 0.1 \cdot N(T) \). As per our given condition this reduced population is equal to the original population of \( 10^{8} \). So, we set up the equation, \( 10^{8} = 0.1 \cdot 10^{8} \cdot 2^{(T/20)} \). If we simplify the equation, we have \( 1 = 0.1 \cdot 2^{(T/20)} \).

Step 3: Solve for T

To solve for \(T\), we can first simplify the equation to \( 2^{(T/20)} = 10 \) by dividing both sides of the equation by \(0.1\). Then, we transform this equation into the logarithmic form \( log_2 (10) = T / 20 \) and, thus, \( T = 20 \cdot log_2 (10) \).

Key Concepts

Population Growth

Understanding population growth in the context of exponential models is crucial when studying the spread of species, particularly bacteria, in biology. Population growth often follows an exponential trend, with the size of the population at any given time being proportionate to its size at a previous time.
For example, the growth of bacteria like E. coli in a laboratory is typically exponential because bacteria divide at regular intervals, provided ample nutrients and space. This means the population can potentially double over regular time periods, such as every 20 minutes in our exercise. If we start with a certain number of bacteria, say, 100, after one interval of 20 minutes, there will be 200; after another 20 minutes, 400; and the pattern continues. An important aspect of exponential growth is the speed at which it increases, which can lead to huge populations in a surprisingly short amount of time.
In our exercise, the doubling effect is impacted by an external factor—a treatment that significantly reduces the population by 90%. However, because of the exponential nature of the growth, the bacteria can return to its original population size swiftly unless the treatment is timed correctly.

Exponential Equations

Exponential equations are mathematical models that describe situations where a quantity grows or decays at a rate proportional to its current value. These equations are commonly used to model population growth, radioactive decay, and interest in finance, among others.
An exponential growth equation generally takes the form of \( N(t) = N_0 \times a^{(t/T)} \), where:

• \( N(t) \) is the population at time \( t \)
• \( N_0 \) is the initial population
• \( a \) is the growth factor
• \( T \) is the doubling time or the time it takes for the population to grow by a certain factor

A key feature of exponential equations is the presence of a base raised to a power. In the context of our exercise, the base is 2, reflecting the doubling nature of bacterial growth, and the power is a function of time emphasizing the change over intervals. The easing of understanding of exponential equations lies in breaking down the components of the formula and analyzing the role each part plays in the growth process. This can be challenging for students unfamiliar with exponential functions, but once grasped, it becomes a powerful tool to predict future growth.

Logarithmic Transformations

Logarithmic transformations are used to solve exponential equations where the variable of interest is in an exponent. This transformation utilizes the logarithm to 'bring down' the exponent, making it easier to solve for the variable.
In the context of our exercise, once the population at time \( T \) before treatment is equal to the original population after a 90% reduction, we arrive at an exponential equation. To solve for \( T \), we apply a logarithmic transformation. By using the logarithm base 2—because our growth is a doubling process—we can transform the equation from \( 2^{(T/20)} = 10 \) to its logarithmic counterpart \( T/20 = \text{log}_2(10) \). This step is crucial as it simplifies the equation to a linear form where \( T \) can be easily isolated and calculated.
Logarithmic transformations are powerful in that they not only help to solve for time periods in population growth but are also prevalent in various scientific and mathematical applications, such as measuring the intensity of earthquakes (Richter scale) or the acidity of solutions (pH scale). Exploring logarithms deepens our ability to understand and manipulate exponential relationships in diverse fields.