Question

10 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 thousandth. 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 hours.

Explanation:

Step1: Input data into calculator

Use a graphing calculator (e.g., TI - 84). Enter the \(x\) - values (0, 1, 2, 3, 4, 5) into list \(L_1\) and the \(y\) - values (250, 330, 580, 800, 1650, 3000) into list \(L_2\).

Step2: Perform exponential regression

On the calculator, go to the STAT → CALC menu. Select ExpReg (exponential regression). The calculator will output values for \(a\) and \(b\) in the equation \(y = a\cdot b^{x}\).
The exponential regression formula is \(y=a\cdot b^{x}\), where \(a\approx229.777\) and \(b\approx1.732\). So the equation is \(y = 229.777\times(1.732)^{x}\).

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

Substitute \(x = 7\) into the equation \(y=229.777\times(1.732)^{7}\).
First, calculate \((1.732)^{7}\approx1.732\times1.732\times1.732\times1.732\times1.732\times1.732\times1.732\approx140.7\).
Then, \(y = 229.777\times140.7\approx32330\).

Answer:

The exponential regression equation is \(y = 229.777\times(1.732)^{x}\). The estimated number of bacteria at the end of 7 hours is 32330.