Question

One hundred bacteria are started in a culture and the number \(N\) of bacteria is counted each hour for 5 hours. The results are shown in the table, where \(t\) is the time in hours. $$ \begin{array}{|l|c|c|c|c|c|c|}\hline t & 0 & 1 & 2 & 3 & 4 & 5 \\\\\hline N & 100 & 126 & 151 & 198 & 243 & 297 \\\\\hline\end{array}$$ (a) Use the regression capabilities of a graphing utility to find an exponential model for the data. (b) Use the model to estimate the time required for the population to quadruple in size.

Short answer

The exponential model is in the form \(N = ab^t\), with \(a\) and \(b\) being constants calculated through regression. Using this model, we solved for \(t\) when \(N = 400\) to estimate the time required for the bacteria population to quadruple.

Step-by-step solution

Recognize the Type of Function

Since the bacteria's growth pattern is not linear but increases rapidly, it suggests an exponential growth. Therefore, an exponential model \(N = ab^t\) will be used, where \(N\) is the number of bacteria, \(t\) is the time in hours, \(a\) is the initial amount (y-intercept when \(t=0\)) and \(b\) is the base (growth factor).

Use the Regression Capabilities of a Graphing Utility

Plug the given data points into a graphing utility to find the exponential regression equation. The equation will be in the form \(N = ab^t\). These tools use least squares regression to minimize the sum of the squares of the residuals (differences between actual and estimated results).

Find Exponential Model

The output equation given by the graphing utility will be your exponential model. This equation can slightly vary depending on the calculator, but it's generally in the form \(N=ab^t\), where \(a\) and \(b\) are constants obtained from the regression.

Estimate the Time Required for the Population to Quadruple

To estimate when the initial population of 100 will quadruple, set \(N = 400\) in the model and solve for \(t\). The solution to this equation will give the time required for the population to quadruple in size.

Key concepts

Exponential Regression

Exponential regression is a type of statistical analysis used to model the relationship between two variables where one variable, typically the dependent variable, changes at an exponential rate relative to the other. In the given exercise, a culture of bacteria demonstrates rapid growth over time, which is best described using an exponential growth model rather than a linear one.

In practice, we use an exponential model of the form \( N = ab^t \) to fit the data. Here, \( N \) is the number of bacteria at time \( t \), \( a \) is the initial amount when \( t=0 \), and \( b \) is the growth factor per time unit. Graphing calculators or statistical software can perform this regression, producing an equation that describes the growth pattern of the bacterial culture based on the available data. By finding the best-fit curve, the method aims to make the sum of the difference between observed values and the values predicted by the model (residuals) as small as possible.

It's crucial for students to understand not only how to plug numbers into a graphing utility but also the underlying principles of exponential functions and the reasons why this type of model is appropriate for the analysis of such growth patterns.

Least Squares Regression

Least squares regression is a traditional approach to finding the best-fitting curve to a set of data points by minimizing the sum of the squares of the residuals, which are the differences between observed and predicted values. It's a widely-used method for both linear and non-linear models, including exponential regression.

In the context of exponential regression, least squares aims to minimize the squared differences between the actual number of bacteria observed at each time point and those predicted by the exponential model \( N = ab^t \). The values of \( a \) and \( b \) are determined such that they provide the 'least squares' solution. Although it might seem complicated at first, modern graphing utilities simplify this process by automatically calculating the coefficients for the best fit equation.

The principles of least squares regression not only support students in understanding statistical analysis but also nurture their ability to critically analyze the validity of models when applied to real-world situations, such as population growth.

Population Growth Estimation

Population growth estimation involves predicting future changes in a population using mathematical models. The exponential growth model is commonly applied when a population grows rapidly, and resources are not limited. In our exercise, we use an exponential growth curve to estimate how long it will take for a population of bacteria to quadruple.

Using the exponential model equation found through regression, we can estimate future population sizes by replacing \( N \) with the desired population size and solving for \( t \). This enables researchers to make informed decisions about growth trends, resource allocation, and planning. For students, understanding this model is essential for predicting how populations, not just of bacteria but any rapidly growing entity, will change over time under ideal growth conditions.

When dealing with homework exercises like these, appreciating the practical applications of these estimations in environmental studies, biology, and resource management can make the learning process more relevant and engaging. Moreover, noting the limitations and assumptions behind these models can help students apply them more effectively, understanding that conditions like limited resources or other inhibiting factors would necessitate different models.