Question

4 the data collected by a biologist showing the growth of a colony of bacteria at the end of each hour are displayed in the table below.

write an exponential regression equation to model these data. round all values to the nearest hundredth. assuming this trend continues, use this equation to estimate, to the nearest ten, the number of bacteria in the colony at the end of 7

Explanation:

Step1: Input data into calculator

Using a graphing calculator or statistical software, input the \(x\)-values (time: \(0,1,2,3,4,5\)) and \(y\)-values (population: \(250,330,580,800,1650,3000\)) for exponential regression.

Step2: Obtain regression equation

The general form of an exponential regression equation is \(y = ab^{x}\). After running the exponential regression, we get \(a\approx229.39\) and \(b\approx1.65\). So the equation is \(y = 229.39(1.65)^{x}\).

Step3: Estimate population at \(x = 7\)

Substitute \(x = 7\) into the equation: \(y=229.39(1.65)^{7}\).
First, calculate \((1.65)^{7}\approx48.77\).
Then, \(y = 229.39\times48.77\approx11188.35\approx11190\) (rounded to the nearest ten).

Answer:

The exponential regression equation is \(y = 229.39(1.65)^{x}\), and the estimated population at the end of 7 hours is \(11190\).