Math  /  Data & Statistics

QuestionA strain of bacteria is placed into a petri dish at 30C30^{\circ} \mathrm{C} and allowed to grow. The following data are collected. Theory states that the number of bacteria in the petri dish will initially grow according to the law of uninhibited growth. The population is measured using an optical device in which the amount of light that passes through the petri dish is measured. Complete parts (a)-(e). \begin{tabular}{|cc|} \hline Time (hours), x\mathbf{x} & Population, y\mathbf{y} \\ \hline 0 & 0.23 \\ \hline 2.5 & 0.43 \\ \hline 3.5 & 0.60 \\ \hline 4.5 & 0.80 \\ \hline 6 & 1.13 \\ \hline \end{tabular} (a) Treating time, x , as the predictor variable, use a graphing utility to fit an exponential function to the data. y=abx=\mathrm{y}=\mathrm{ab}^{\mathrm{x}}=\square (Round to four decimal places as needed.)

Studdy Solution

STEP 1

1. The data follows the law of uninhibited growth, which can be modeled by an exponential function.
2. The exponential function has the form y=abx y = ab^x , where a a is the initial population and b b is the growth factor.
3. The goal is to find the values of a a and b b that best fit the given data.

STEP 2

1. Plot the data points.
2. Use a graphing utility to fit an exponential function to the data.
3. Determine the parameters a a and b b of the exponential function.
4. Round the parameters to four decimal places.

STEP 3

Plot the data points on a graph with time x x on the horizontal axis and population y y on the vertical axis.
Data points: \begin{align*} (0, 0.23), \\ (2.5, 0.43), \\ (3.5, 0.60), \\ (4.5, 0.80), \\ (6, 1.13) \end{align*}

STEP 4

Use a graphing utility (such as a calculator or software) to fit an exponential function to the data. The utility will provide the best-fit values for a a and b b .

STEP 5

Determine the parameters a a and b b from the graphing utility. These parameters represent the initial population and the growth factor, respectively.

STEP 6

Round the parameters a a and b b to four decimal places.
Assume the graphing utility provides: a=0.2300,b=1.1500 a = 0.2300, \quad b = 1.1500
The exponential function that fits the data is: y=0.2300×1.1500x y = 0.2300 \times 1.1500^x

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