A hive of bees contains 27 bees when it is first discovered. After 3 days, there are 36 bees. It is determined that the population of bees increases exponentially.How many bees are will there be after 30 days?

Accepted Solution

Answer:After 30 days, there will be around 542 bees in the hive.Step-by-step explanation:GivensThe hive contains 27 bees in first place.After 3 days, there are 36 bees.The population growth is modelled by the expression[tex]A=A_{0}e^{kt}[/tex]Where [tex]A[/tex] is the population after [tex]t[/tex] days, [tex]A_{0}[/tex] is the initial population, [tex]t[/tex] is days and [tex]k[/tex] is the constant of proportionality.Basically, in these kind of problems, we use the given information to find [tex]k[/tex] first[tex]A=A_{0}e^{kt}\\36=27e^{3k}\\\frac{36}{27}= e^{3k}\\ln(\frac{4}{3})=ln(e^{3k})\\3k=ln(\frac{4}{3})\\k\approx 0.10[/tex]Now, with this constant, we find the population of bees after 30 days.[tex]A=A_{0}e^{kt}\\A=27e^{0.10(30)}\\A=27e^{3}\approx 542[/tex]Therefore, after 30 days, there will be around 542 bees in the hive.