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Is a linear model appropriate? The number, in thousands, of bacteria in a petri dish is given by the table below. Time is measured in hours. $$ \begin{array}{|c|c|} \hline \begin{array}{l} \text { Time in hours since } \\ \text { experiment began } \end{array} & \begin{array}{c} \text { Number of bacteria } \\ \text { in thousands } \end{array} \\ \hline 0 & 1.2 \\ \hline 1 & 2.4 \\ \hline 2 & 4.8 \\ \hline 3 & 9.6 \\ \hline 4 & 19.2 \\ \hline 5 & 38.4 \\ \hline 6 & 76.8 \\ \hline \end{array} $$ The table below shows enrollment, \({ }^{18}\) in millions of people, in public colleges in the United States during the years from 2001 through 2005 . $$ \begin{array}{|c|c|} \hline \text { Date } & \text { Enrollment in millions } \\ \hline 2001 & 12.23 \\ \hline 2002 & 12.75 \\ \hline 2003 & 12.86 \\ \hline 2004 & 12.98 \\ \hline 2005 & 13.02 \\ \hline \end{array} $$ a. Plot the data points for number of bacteria. Does it look reasonable to approximate these data with a straight line? b. Plot the data points for college enrollment. Does it look reasonable to approximate these data with a straight line?

Step by step solution

01

Plotting Number of Bacteria Data Points

To address part (a), plot the time (in hours) on the x-axis and the number of bacteria (in thousands) on the y-axis. The data points are: (0, 1.2), (1, 2.4), (2, 4.8), (3, 9.6), (4, 19.2), (5, 38.4), (6, 76.8). Notice that the number of bacteria doubles each hour, indicating an exponential growth pattern. Therefore, it is not reasonable to approximate these data with a straight line, as the growth is not linear but exponential.

02

Plotting College Enrollment Data Points

To address part (b), plot the year (2001 to 2005 as x-values) on the x-axis and the enrollment in millions on the y-axis. Mark the data points: (2001, 12.23), (2002, 12.75), (2003, 12.86), (2004, 12.98), (2005, 13.02). These points form a nearly straight line, indicating a consistent, linear increase in enrollment each year. Therefore, it is reasonable to approximate these data with a straight line.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Growth

Exponential growth is a rapid increase in quantity where the rate of growth becomes quicker over time. In the given exercise, we observe this concept through the petri dish bacteria experiment, where the number of bacteria doubles every hour.
This mean the growth pattern is exponential, not linear. When looking at the initial data points:

• At 0 hours, there are 1.2 thousand bacteria.
• After 1 hour, it doubles to 2.4 thousand.
• This pattern of doubling continues in a consistent manner.

Since exponential growth involves a multiplying factor, it significantly deviates from a straight-line trend, which would indicate constant growth. In mathematical terms, it resembles the expression: \[ P = P_0 \cdot (1 + r)^t \] where \(P\) is the population after time \(t\), \(P_0\) is the initial population, and \(r\) represents the rate of growth. As \(t\) increases, the population increases at an accelerating rate, leading to a rapid escalation in numbers.Understanding exponential growth is crucial for situations involving rapid change, such as population growth, radioactive decay, and certain financial scenarios.

Data Analysis

Data analysis is the process of evaluating data with the goal of uncovering valuable insights or identifying patterns in the data set. In our current context, we're evaluating two different data sets to determine their growth patterns. The first set involves bacteria growth in a petri dish, which we've identified as exponential. The second set involves college enrollment over a span of five years. When analyzing data:

• Look for trends or patterns in the numbers.
• Assess whether changes are consistent (linear) or disproportionate (exponential).
• Use visualization tools, like graphs, to better understand the data.

In this exercise, we plotted the data points on graphs to analyze the patterns.
For the bacteria, the rate of increase highlighted the exponential nature. For college enrollment, the relatively small and consistent increase in numbers suggested a linear pattern. Conducting data analysis not only involves plotting data but also interpreting results to make informed conclusions about potential growth trends and interventions.

Graph Interpretation

Graph interpretation involves understanding and making sense of the visual representation of data on a graph. In the given exercise, we plotted data for both bacteria growth and college enrollment to examine their respective trends. For the bacteria data:

• Plotted with time on the x-axis and number of bacteria on the y-axis.
• Showed a steep, upwards-accelerating curve—indicative of exponential growth.

For the college enrollment data:

• Plotted with year on the x-axis and enrollment numbers on the y-axis.
• Displayed a gradual slope, which suggests a linear trend.

When interpreting graphs, key points include:

• Checking the axes to understand what variables you're dealing with.
• Noting the general shape of the plotted data—whether it suggests a linear or exponential relationship.
• Drawing conclusions based on trends and identifying if further investigation is necessary.

Graphs serve as a powerful tool to visualize data, helping to make complex datasets more accessible and highlighting key trends at a glance.