Problem 15 - Differential Equations and Linear Algebra

Question

If the size \(P(t)\) of a culture of bacteria (measured in thousands of bacteria) is measured at various time intervals (measured in minutes) with the data in the table below, use least squares to estimate \(P(t)\) at any time \(t:\) $$\begin{array}{c|c|c|c|c} \hline \text { time } t \text { (in minutes) } & 0 & 12 & 20 & 24 \\ \hline \text { number of bacteria } & 1.5 & 5 & 11 & 16 \\ P(t) \text { (in thousands) } & & & & \\ \hline \end{array}$$

Step-by-Step Solution

Answer

In summary, using the least squares method, we found the best-fit line equation to estimate the size of the bacteria culture at any time interval \(t\) as \(P(t) = \frac{587}{42}t - \frac{191}{42}\). To estimate the size of the culture for a specific time interval, simply substitute the desired value of \(t\) into the equation.

Step 1: Organize the date in a table

Begin by organizing the given data points in a table with columns for the time elapsed in minutes (\(x\)), the number of bacteria (\(y\)), \(x^2\), \(y^2\), and \(xy\), and calculate the product of every \(x\) and \(y\); every \(x^2\); and every \(y^2\). \[ \begin{array}{c|c|c|c|c|c} \hline x & y & x^2 & y^2 & xy \\\ \hline 0 & 1.5 & 0 & 2.25 & 0 \\\ 12 & 5 & 144 & 25 & 60 \\\ 20 & 11 & 400 & 121 & 220 \\\ 24 & 16 & 576 & 256 & 384 \\\ \hline \sum{x} = 56 & \sum{y} = 33.5 & \sum{x^2} = 1120 & \sum{y^2} = 404.25 & \sum{xy} = 664 \\\ \hline \end{array} \]

Step 2: Calculate the mean values of \(x\) and \(y\)

Calculate the mean values of \(x\) and \(y\) by dividing the sum of their respective values by the number of data points, which is 4. \[\bar{x} = \frac{\sum{x}}{n} = \frac{56}{4} = 14\] \[\bar{y} = \frac{\sum{y}}{n} = \frac{33.5}{4} = 8.375\]

Step 3: Calculate the slope and the intercept of the best-fit line

Use the formulas for the slope (m) and the intercept (b) of the best-fit line: \[m = \frac{n\sum{xy} - \sum{x}\sum{y}}{n\sum{x^2} - (\sum{x})^2}\] \[b = \frac{\sum{y} - m\sum{x}}{n}\] Substitute the given values from our table and mean calculations: \[m = \frac{4(664) - 56(33.5)}{4(1120) - (56)^2} = \frac{587}{42}\] \[b = \frac{33.5 - (\frac{587}{42})(56)}{4} = -\frac{191}{42}\]

Step 4: Write the least squares equation and find \(P(t)\)

With the slope and intercept, write the least squares equation in the form of \(P(t) = mt + b\). We can now find the estimated size \(P(t)\) at any time \(t\). \[P(t) = \frac{587}{42}t - \frac{191}{42}\] Now, you can plug in any time \(t\) to estimate the size of the culture at that time.

Key Concepts

Bacterial Growth

Understanding bacterial growth is essential in microbiology and many related fields. Bacterial growth refers to the increase in the number of bacteria in a population rather than an individual increase in size. This growth typically follows a predictable pattern under controlled conditions, composed of four stages:

• **Lag Phase**: The bacteria acclimate to their environment. Not much increase in number occurs, as they are preparing for growth.
• **Exponential (Log) Phase**: The bacteria begin to divide at a consistent rate, resulting in the exponential growth of the population.
• **Stationary Phase**: Nutrient depletion and waste accumulation slow growth, and the rate of division equals the rate of death.
• **Death Phase**: Nutrients are exhausted, and waste products accumulate, leading to a decrease in the population.

In the context of the exercise provided, the data points in the table reflect measurements from the exponential growth phase, where the population is rapidly increasing. This phase is critical in data analysis as it provides insights into the growth dynamics of bacterial populations, which can be modeled effectively using mathematical approaches like least squares regression.

Data Analysis

Data analysis involves systematically applying statistical and logical techniques to describe and evaluate data. In this exercise, data analysis is used to process and interpret the measurements of bacteria growth over time. The goal is to identify meaningful patterns and trends from the data provided.

The data analysis steps begin with organizing the data in a structured format—such as a table—where calculations like sums, squares, and products can be easily performed. Summing the data gives insights into overall trends, while calculating products and squares helps in fitting a regression line. It's crucial to interpret these values correctly to make accurate predictions and conclusions. Data analysis provides the foundation for applying statistical models that describe the relationship between variables—in this case, time and bacterial count. Understanding these relationships helps in developing predictive models, essential in various biological and environmental applications.

Linear Regression

Linear regression is a statistical method used to model the relationship between two variables by fitting a linear equation to observed data. In the exercise provided, linear regression is applied to estimate the growth of a bacterial culture over time using least squares regression. The least squares method minimizes the sum of the squares of the differences between observed and predicted values, thus providing the best-fit line.

To perform linear regression:

• Calculate the means of the independent ( x ) and dependent ( y ) variables.
• Determine the slope ( m ) and intercept ( b ) of the line using specific formulas. These capture the changes in response (bacterial count) with time.
• Formulate the linear equation: y = mx + b , which can now be used to make predictions.

The primary goal is to predict the dependent variable's value, given an independent variable. In the context of bacterial growth, this allows us to estimate the population size at any future time, facilitating planning and decision-making in research or industrial applications.