A researcher is trying to determine the doubling time for a...

A researcher is trying to determine the doubling time for a population of the bacterium Giardia lamblia. He starts a culture in a nutrient solution and estimates the bacteria count every four hours. His data are shown in the table. $$\begin{array}{|c|c|c|c|c|c|c|c|}\hline \text { Time (hours) } & {0} & {4} & {8} & {12} & {16} & {20} & {24} \\ \hline \text { Bacteria count } & {37} & {47} & {63} & {78} & {105} & {130} & {173} \\ \hline\end{array}$$ (a) Make a scatter plot of the data. (b) Use a graphing calculator to find an exponential curve $f(t)=a \cdot b^{t}$ that models the bacteria population $t$ hours later. (c) Graph the model from part (b) together with the scatter plot in part (a). Use the TRACE feature to determine how long it takes for the bacteria count to double.

A researcher is trying to determine the doubling time for
a population of the bacterium Giardia lamblia. He starts
a culture in a nutrient solution and estimates the bacteria
count every four hours. His data are shown in the table.
$$\begin{array}{|c|c|c|c|c|c|c|c|}\hline \text { Time (hours) } & {0} & {4} & {8} & {12} & {16} & {20} & {24} \\ \hline \text { Bacteria count } & {37} & {47} & {63} & {78} & {105} & {130} & {173} \\ \hline\end{array}$$
(a) Make a scatter plot of the data.
(b) Use a graphing calculator to find an exponential curve
$f(t)=a \cdot b^{t}$ that models the bacteria population
$t$ hours later.
(c) Graph the model from part (b) together with the scatter
plot in part (a). Use the TRACE feature to determine
how long it takes for the bacteria count to double.

Key Concepts

Exponential Growth

Exponential growth describes a process where the rate at which a quantity increases is proportional to its current size. This results in a rapid escalation over time and is typically modeled by functions of the form f(t) = a · b^t, where 'a' represents the initial amount and 'b' is the growth factor. This concept is fundamental in fields like biology, finance, and physics.

Scatter Plot

A scatter plot is a graphical representation used to display the relationship between two variables by plotting data points on a two-dimensional coordinate system. It is especially useful for visualizing the spread and trend of data, identifying correlations, and checking the fit of a mathematical model to observed values.

Exponential Regression

Exponential regression is a statistical method used to fit an exponential model to a set of data. The process determines the constants in the model f(t) = a · b^t that best capture the underlying trend of the data, allowing for the prediction of future values in exponentially growing systems.

Doubling Time

Doubling time is the interval required for a quantity undergoing exponential growth to double in size. It is an important metric in understanding growth dynamics, enabling the estimation of how quickly a population or investment will expand, and is determined by evaluating the exponential function at twice its initial value.

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