The population (in thousands) of two different types of insects on an island can be modelled by the following functions: Butterflies: $f\left(t\right)=a+b\sin\left(mt\right)$f(t)=a+bsin(mt), Crickets: $g\left(t\right)=c-d\sin\left(kt\right)$g(t)=c−dsin(kt)
$t$t is the number of years from when the populations started being measured, and $a$a,$b$b,$c$c,$d$d,$m$m, and $k$k are positive constants. The graphs of $f$f and $g$g for the first $2$2 years are shown below.
State the function $f\left(t\right)$f(t) that models the population of Butterflies over $t$t years.
State the function $g\left(t\right)$g(t) that models the population of Crickets over $t$t years.
How many times over a $18$18 year period will the population of Crickets reach its maximum value?
How many years after the population of Crickets first starts to increase, does it reach the same population as the Butterflies?
Solve for $t$t, the number of years it takes for the population of Butterflies to first reach $200000$200000.
Display and interpret the graphs of functions with the graphs of their inverse and/or reciprocal functions