A researcher observes the population of a bacterial culture
MATHEMATICS · Multi-part · AP · AP Precalculus · Exponential and Logarithmic Functions · Exponential and Logarithmic Functions
Problem
A researcher observes the population P(t) of a bacterial culture, where t is measured in hours. The population is hypothesized to satisfy P(0)=P0 and to change over time according to the model P(t)=P0ekt for some constant k>0. The researcher collects population counts at several time values and wishes to evaluate whether the model is reasonable and, if so, to estimate the parameters.
(a) Show algebraically that if the proposed model is valid, a plot of lnP versus t should be a straight line. Identify the slope and y-intercept of that line in terms of k and P0.
(b) Describe a procedure—without performing any numerical calculations—that the researcher could follow to use the experimental data and the linear relationship found in part (a) to obtain an estimate of k. Your answer should reference specific graphical or statistical features that would be examined.
(c) Suppose statistical software applied to the data returns the regression equation lnP=1.20+0.35t. Interpret both the slope 0.35 and the intercept 1.20 in the context of bacterial growth, including appropriate units.
(d) Explain how an examination of the residuals for the linear fit in part (c) would help assess the validity of the exponential growth model for this culture.