Chapter 7.4, Problem 1E

To determine

To find : the population size after six days

Answer to Problem 1E

The population size is $P=234.991$

Explanation of Solution

Given information : The question is

A population of protozoa develops with a constant relative growth rate of $0.7944$ per member per day. On day the zero population of two member, So,

  $\frac{dP}{dt}=0.7944$

Calculation : Setup a differential equation using the information provided in the question

Let $P$ be the population size and $t$ be time in hours

  $\begin{array}{l}\frac{dP}{dt}=0.7944P\\ dP=0.7944P\times dt\\ \frac{dP}{P}=0.7944dt\end{array}$

Separate the variables

  $\begin{array}{l}{\displaystyle \int \frac{dP}{P}={\displaystyle \int 0.7944dt}}\\ \ln \left|P\right|=0.7944t+C\\ \text{integrate both sides}\\ \left|P\right|=e^{0.7944t+C}\\ \left|P\right|=e^C\times e^{0.7944t}\\ P=C{e}^{0.7944t}\left(\text{raise to the }{e}^x\text{ power}\right)\\ 2=C{e}^{0.7944\times 0}\\ 2=C\times 1\\ 2=C\\ P=2\times e^{0.7944\times 6}\\ P=234.991\end{array}$