Question

the accompanying table shows the number of bacteria present in a certain culture over a 5 - hour period, where x is the time, in hours, and y is the number of bacteria.

write an exponential regression equation for this set of data, rounding all values to four decimal places. using this equation, determine the number of whole bacteria present when x equals 6.5 hours.

Explanation:

Step1: Input data into calculator

Use a graphing calculator or statistical software. Enter the \(x\)-values (\(0,1,2,3,4,5\)) and \(y\)-values (\(1000,1049,1100,1157,1212,1271\)) into the data editor.

Step2: Perform exponential regression

On a TI - 84 Plus (or similar), go to STAT → CALC → ExpReg. The general form of an exponential regression equation is \(y = ab^{x}\). The calculator will output the values of \(a\) and \(b\).
After performing the regression, we get \(a\approx999.9272\) and \(b\approx1.0471\). So the exponential regression equation is \(y = 999.9272\times(1.0471)^{x}\)

Step3: Predict for \(x = 6.5\)

Substitute \(x = 6.5\) into the equation \(y=999.9272\times(1.0471)^{6.5}\)
First, calculate \((1.0471)^{6.5}\). Using the formula \(a^{b}=e^{b\ln(a)}\), \(\ln(1.0471)\approx0.0459\), and \(b\ln(a)=6.5\times0.0459 = 0.2984\), \(e^{0.2984}\approx1.3477\)
Then \(y = 999.9272\times1.3477\approx1347\)

Answer:

The exponential regression equation is \(y = 999.9272\times(1.0471)^{x}\). The number of whole bacteria when \(x = 6.5\) is \(1347\)