6.04 Problem solving with trigonometric functions
Interactive practice questions
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.
Three objects, $X$X, $Y$Y and $Z$Z are placed in a magnetic field such that object $X$X is $2$2 cm from object $Y$Y and $4$4 cm from object $Z$Z. As object $X$X is moved closer to line $YZ$YZ, object $Y$Y and $Z$Z move in such a way that the lengths $XY$XY and $XZ$XZ remain fixed.
Let $\theta$θ be the angle between sides $XY$XY and $XZ$XZ, and let the area of triangle $XYZ$XYZ be represented by $A$A.
The change in voltage of overhead power lines over time can be modelled by a periodic function that oscillates between $-300$−300 and $300$300 kiloVolts with a frequency of $40$40 cycles per second.