In a microbiology experiment

MATHEMATICS · Multi-part · AP · AP Precalculus · Exponential and Logarithmic Functions · Exponential and Logarithmic Functions

Problem

In a microbiology experiment, the population of bacteria (measured in thousands of cells) is modeled for

t>0

hours by

P(t)=A t^{r},

where

A>0

and

r>0

are constants. Researchers collect the following information at

t=2

hours: • The population size is

P(2)=500

(thousand cells). • The instantaneous growth rate is

P'(2)=650

(thousand cells per hour). Parts: (a) Write two separate equations involving

A

and

r

that arise from the population size and growth-rate data at

t=2

. (b) Solve the system you obtained in part (a) to determine the exact value of

r

. Clearly justify each algebraic step, including any use of logarithms. (c) Using the value of

r

from part (b), determine the constant

A

. (d) Estimate the population at

t=3

hours. Clearly indicate units and comment on whether the growth appears to be accelerating or decelerating at

t=3