Answer :
Answer:
The population size after 7 days is about 427.
Step-by-step explanation:
If [tex]y(t)[/tex] is the value of a quantity [tex]y[/tex] at time [tex]t[/tex] and the if the rate of change of [tex]y[/tex] with respect to [tex]t[/tex] is proportional to its size [tex]y(t)[/tex] at any time, then
[tex]\frac{dy}{dt} =ky[/tex]
where [tex]k[/tex] is a constant.
This equation is sometimes called the law of natural growth (if [tex]k>0[/tex]).
The only solutions of the differential equation [tex]\frac{dy}{dt} =ky[/tex] are the exponential functions
[tex]y(t)=y(0)e^{kt}[/tex]
Let [tex]P[/tex] be the population size and let [tex]t[/tex] be the time variable, measured in hours. Since the relative growth rate is 0.6671, then the differential equation that models this growth is
[tex]\frac{dP}{dt} =0.6671\cdot P[/tex]
According with the above information the solution to this differential equation is
[tex]P(t)=P(0)e^{0.6671t}[/tex]
On day zero the population consists of 4 members [tex]P(0)=4[/tex].
Therefore, the population size after 7 days is
[tex]P(7)=4e^{0.6671\cdot 7}\\\\P(7)=4e^{4.6697}=4\cdot \:106.66573=426.66295[/tex]
about 427.