Trig Functions and Graphs (radians)

Hong Kong

Stage 4 - Stage 5

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`+`b``s``i``n`(`m``t`), Crickets: $g\left(t\right)=c-d\sin\left(kt\right)$`g`(`t`)=`c`−`d``s``i``n`(`k``t`)

$t$`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.

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a

State the function $f\left(t\right)$`f`(`t`) that models the population of Butterflies over $t$`t` years.

b

State the function $g\left(t\right)$`g`(`t`) that models the population of Crickets over $t$`t` years.

c

How many times over a $18$18 year period will the population of Crickets reach its maximum value?

d

How many years after the population of Crickets first starts to increase, does it reach the same population as the Butterflies?

e

Solve for $t$`t`, the number of years it takes for the population of Butterflies to first reach $200000$200000.

Easy

Approx 19 minutes

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 $YZ$`Y``Z`, object $Y$`Y` and $Z$`Z` move in such a way that the lengths $XY$`X``Y` and $XZ$`X``Z` remain fixed.

Let $\theta$`θ` be the angle between sides $XY$`X``Y` and $XZ$`X``Z`, and let the area of triangle $XYZ$`X``Y``Z` be represented by $A$`A`.

A metronome is a device used to help keep the beat consistent when playing a musical instrument. It swings back and forth between its end points, just like a pendulum.

For a particular speed, the given graph represents the metronome's distance, $x$`x`cm, from the centre of its swing, $t$`t` seconds after it starts swinging. Negative values of $x$`x` represent swinging to the left, and positive values of $x$`x` represent swinging to the right of the centre.

Sounds around us create pressure waves. Our ears interpret the amplitude and frequency of these waves to make sense of the sounds.

A speaker is set to create a single tone, and the graph below shows how the pressure intensity ($I$`I`) of the tone, relative to atmospheric pressure, changes over $t$`t` seconds.

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