Mathematics

Problem

The size of a bacterial population in a laboratory culture is given by

P(t)=2000e^{0.04t},

where

P(t)

denotes the number of bacteria present

t

hours after the start of the experiment. a. (i) Obtain a formula for the instantaneous growth rate

\dfrac{dP}{dt}

. b. (ii) Compute the instantaneous growth rate at

t=5\,\text{h}

. c. (iii) Find the average growth rate of the population over the interval

0\le t\le5\,\text{h}

. d. (iv) Determine the time

t

(to two decimal places) at which the instantaneous growth rate first reaches

300

bacteria per hour.