In an ecology experiment, a number of mosquitoes are placed in a containex9 js-,ul lonn,ka 6zn ,ykb:7r with water and vegetation. The population of mosquitoes, P, increases and can be modelled by the functi -n,ou7,n xsl yn: a,j9zbk6lkon
P(t)=24$\times4^{0.385t}$,t≥0,
where t is the time, in days, since the mosquitoes were places in the container.
1.Find the number of mosquitoes:
(1)initially placed in the container; P =   
(2)in the container after 55 days. P ≈   
The maximum capacity of the container is 50005000 mosquitoes.
2.Find the number of days until the container reaches its maximum capacity. t ≈   

  The amount of water, W, in litres, remaining in a cooking pwnu (jhm5o) np) y9n(-kkl +rdot after it is placed onto a hy9+n)dphw n o(ul kmn5(r) k-jot stove is given by the function
W(t)=40$(0.75)^t$,t≥0
where t is the time in hours after the pot is placed on the stove.
1.Find
(1)the initial amount of water in the pot. W =    litres
(2)the percentage of water that leaves the pot in the form of steam each hour. W =    litres
2.Calculate the amount of water remaining in the pot after 55 hours. W =    litres

  An ice cube was dropped into a cup of warm w-v*o(qz,c:lwn gvo),m7ukbvkh t y359 jnnw9ater. The weight of the ice cube, W, in grams, reduces as the ice cube dissolves in the water. The weight can be modelled by the following function, W(t), where t is the time in - ,9y,n9(qo3:v7 ck tvkbjl5*h)ugwmzovn nw seconds after the dissolving starts.
W(t)=28$(0.98)^t$,t≥0
1.Find
(1)the initial weight of the ice cube. W =    g
(2)the percentage of the ice cube dissolving every 55 seconds.    %
2.Calculate the weight of the ice cube remaining after one minute. W =    g

  Oliver throws a coin from the open window onto flat horizontal7c*j cub e(vua-7/h4noo nr)b ground in front of his office building. ho*ujevan4(-n)r 7u /cob b 7c
The path of the coin is modelled by the quadratic curve y=8+3x−0.5x2 , where $x$ represents the horizontal distance the coin is thrown and $y$ represents the height of the coin above the ground. All distances are measured in metres. The wall of the building lies along the $y$-axis. The coin starts its flight at point A, reaches its maximum height at point B, and lands on the ground at point C, as shown on the diagram below.

1.Write down the height in metres from which the coin was thrown.    m
2.Calculate the maximum height, above the ground, reached by the coin.    m
3.Find the horizontal distance from the base of the wall to the point at which the coin hits the ground.    m

  After 1960, the economy growth in Australia has grown in an exponential fashi5 )s2w hjlvur,on. Australian GDP per capita, G(t), in United,2jsvwr) 5 ulh States dollars (USD) is modelled by the function
G(t)=1806$\times(1.037)^t$,
where t is the number of years since 1960, and t≥0.
1.Write down, in USD, Australian GDP per capita in 19601960.    USD
2.Find Australian GDP per capita in 1961. ≈    USD
3.Find the year for which Australian GDP per capita reaches 100000100000 USD. Give your answer correct to the nearest year. t ≈   

  A population of 50 hamsters was introduced j8x- oj*:5j0 qy: 2qbppngowpto a new town. One month later, the number of hamsters was 6262. The number of hamsters, P, can be modwp5j:-x 8oq*pnpo2 :jg q0jybelled by the function
P(t)=50$\times\,b^t$,t≥0,
where t is the time, in months, since the hamsters were introduced to the town.
1.Find the value of b. b =   
2.Calculate the number of hamsters in the town after 66 months.
A wildlife specialist estimates that the town has enough drink and food to support a maximum population of 20002000 hamsters. P =   
3.Calculate the number of months it takes for the hamster population to reach this maximum. t =    months

  The sound intensity level, D, in decibels (dB),k/dh 5qce. zh2uc;-d(iw g *ml can be modelled by the function .2cdzqh*w(kg -;l5h i/u cde m
D(I)=10$log_{10}​$(I)+120,I≥0,
where I is the sound intensity, in watts per square meter (W /$m^2$).
1.A vacuum cleaner has a sound intensity of 1.6$\times10^{−5}$ W /$m^2$. Calculate the intensity level of the vacuum cleaner. D =    dB
2.A fire truck siren has an intensity level of 124 dB. Find the sound intensity of the siren. I ≈    W /$m^2$

  The average daily hours ofzgkj -4s ( i3gnhw1 76ebi6q5bojjna; sunlight, S, during a 12-month period in a particular6sj1 6eikgqh 4 3obj7jw-(5ib nznga ; town is given by the function
$S(t)=\,acos(bt−π)+d$,
where time,t, is measured in months, a and d are constants, and b is measured in degrees.
The graph of $S$ versus t is shown in the diagram below.


1.Find the value of
(1)a =   
(2)d =   
2.Find the value of b.b =    $^{\circ}C$
3.Write down the values of t when the average daily sunlight is 10 hours.

  The diagram below shows the graph o he;4kqs iz8 (0btgzh y7x8/qpf a quadratic function f(x)=$2x^2$+bx+c.


1.Write down the value of c. c =   
2.Find the value of b and write down f(x). f(x) =$ax^b+cx+d$ a =    b =    c =    d =   
4.Calculate the coordinates of the vertex of the graph of f. f (-1) = (a,b) a =    b =   

  Atmospheric pressure, P, in kPa, decreases exponentially wit40bru iwh0ud/ h increasing height above wih4u u00rd b/sea level, $h$. The atmospheric pressure can be modelled by the function
$P(h)$​=101$\times(\frac{25}{22}​)^{-h}$,
​where ℎh is the height above sea level in kilometres.
1.Write down the exact atmospheric pressure at sea level, in kPa.
Mount Kosciuszko is the highest mountain in Australia with a height of 2228 metresmetres above sea level at the top.    kPa
2.Calculate the atmospheric pressure at the top of the Mount Kosciuszko. P ≈   
3.Calculate the height where the atmospheric pressure is equal to 10 kPa. h =   

  The size of the trunk diameter of a chest a.k2qzo;j (dqksov75i l)g5c nut oak tree, $d$, in centimetres, can be modelled by the function
$d(t)=9ln(0.25t)$,t≥5,where $t$ is the age, in years, of the tree.
1.Find the size of the trunk diameter of a 10-year-old chestnut oak tree. d =   
2.The size of the trunk diameter of a chestnut oak tree is 15 cm. Find its age. t =   
A chestnut oak tree can be harvested for wood when the size of its trunk diameter reaches 20 cm.
3.Find the age of the chestnut oak tree when it is old enough to be harvested for timber. t =   

  The magnitude of an e-2shsol*pu;w ap v8 1rarthquake, $M$, on the Richter scale, can be modelled by the function
M(E)=$\frac{2}{3}​log_{10}​(E)−3.2$,where E is the amount of energy, in joules, released by the earthquake.
1.Find the magnitude of an earthquake which releases $6.3\times10^{13}$ joules of energyenergy. M ≈   
The Great Chilean Earthquake of 1960, the most powerful earthquake recorded in the 2020th century, had a magnitude of 9.5 on the Richter scale.
2.Find the amount of energy, in joules, released by the Great Chilean EarthquakeEarthquake. E ≈    $\times10^{19}$

  The average time, T, in seconds, it takes for sf)6n-3 ak ljea customer to find their favourite menu item in an alphabetical menu list c eln) 6a-3kjfsan be modelled by the function
$T(n)​=\frac{3}{2}​log​_{(n+1)},n≥1$,
​where $n$ is the number of different menu items in the list.
Mehmet visits Taste of Adana Restaurant for the first time to have a lunch.
1.The restaurant serves 10 different fresh salads. Find the time it takes for Mehmet to find Persian Salad in the Fresh Salads section of the menu list. T ≈    seconds
2.It takes 7.5 seconds for Mehmet to find Lamb Chops Kebab in the Adana Kebab section of the menu list. Find the number of different kebab dishes served by the restaurant. n =   

The water depth, D, in metres, in a harbour on a particular day can be mod.*;xylcz;b* ww 939pdppk uhkelled by *uk;.9hywp 3 bk*;wcpd 9 lzxpthe equation
D=2.5cos($30^{\circ}$$\times{t}$)+5,0≤t≤24,where t is the elapsed time, in hours, since midnight.
1.Draw the graph of D versus t on the grid below.



2.Find the lowest and highest depths of water in the harbour, and the times when they occur.
3.A large yacht has a draught of 3 metres. Decide whether the yacht will be able to enter the harbour at 5 pm.

The water depth, $D$, in metres, in a harbour on a particular can be modelled by the equation
D=3sin(30$^{\circ}$×t)+5,0≤t≤24,
where $t$ is the elapsed time, in hours, since midnight.
Draw the graph of $D$ versus $t$ on the grid below.Find the lowest and highest depths of water in the harbour, and the times when they occur.
A large boat has a draught of 2.5 metres. Decide whether the boat will be able to enter the harbour at 8 pm.

  The tip of a large wind turbine's blade rotates at a consta.fqbkx/fg6 vxn j77hlr 16zo( nt speed on a windy d.q 7z h/kofjx7xrbgv n(66 1flay. Its height, $H$ , in metres, above the ground is modelled by the function
H(t)=52sin(60$^{\circ}$$\times$t)+80,where $t$ is the elapsed time, in seconds, since the turbine was accelerated to its max speed.
1.Write down the minimum height of the blade's tip above the ground. $H_{min}$ =    m
2.Find the height of the blade's tip above the ground after 88 seconds. H =    m
3.Find the time it takes for the blade's tip to complete one full revolution.

  PM2.5 particles are tiny air p+ l/tdyo*0s4p 2m-lqg om 2qlmollutants with a diameter of 2.5 micrometres or less. In comparison, the average diameter of a human hair is about 70 micrometresmicrometres. Air pollution from diesel & petrol engines form part of PM2.5 emissions and of particularparticular concern due to the healolmt m-l+2gm0* sp yql4/2doqth impacts.

During a particular working day, the rate of PM2.52.5 emissions, in grams per second, from all road vehicles in Beijing, China is modelled by the function
$E(t)=6sin(\frac{π}{6}t−\frac{5π}{6}​)+9,0$≤t≤24,
where $t$ is the elapsed time, in hours, since midnight.
1.Find the maximum rate of PM2.5 emissions during the day in Beijing, China. $E_{max}$ =    g/sec
2.Find the times at which the maximum rate of PM2.5 emissions occur during the day. t =      

  For 2019, the income tax rates for a resident in UK are shown on the gra+lt fy3oj44wc ph belowfjo4wc4lt +y3.

1.Write down the highest amount of income that is free of tax.   
2.Calculate the amount of tax payable on the first $£50000$ of income. Tax payable =   
3.Theresa earns $£148500$ in 2019. Find the amount of her income tax payable. Tax payable =   

  The orbital velocity, v, of a spaceb156/kfwztcx2tcpa + ao.2u c*uc h8acraft in km/s is given by
$v=−0.0098t+clnR$,where t is the firing time in seconds, c is the exhaust velocity in km/s and R is the mass ratio.
If R=30, find
1.The orbital velocity, v, after a firing time of 150 seconds and an exhaust velocity of 3 km/s. v ≈    km/s
2.The exhaust velocity, c, of the spacecraft traveling at an orbital velocity of 9.5 km/s after a firing time of 150 seconds. c ≈    km/s

  The total cost of a gym membership, C, in Aje;34 w)3dbn :mvr dm4hz gqe)ustralian dollars (AUD), in John's Gym can be mode );3r ejqgnz:m4b)edmh 4wv3dlled by the function
C=65t+30,​
where t is the time in months.
1.Calculate the gym membership cost over a 66 month duration. C =    AUD
2.On the grid below, sketch the graph of the function C=65t+30, for t≥0.

3.Calculate the time, t, in months, when the total cost reach 290 AUD.
In the neighbouring Jetts Gym, the initial payment is 75dollars higher than in John's Gym, however the monthly fee is lower at 30dollars per month. t =    months
4.Find the number of months it takes for the total cost to be less by attending Jetts Gym in comparison to John's Gym.

  The total cost of a ride, C, in British pounds (GBP), oeo zddf6w.(7, ksa5vr n BlackCabs can be modelledmodelled byosredf6.za7 kw,v(5d the function
C=1.75x+3.00,where x is the distance travelled in kilometres.
1.Calculate the total cost of an 88 kilometre ride in a BlackCab. C =    GBP
2.On the grid below, sketch the graph of the function C=1.75x+3.00, for x≥0.

3.Calculate the distance travelled, x, in kilometres, when the total cost of a ride reach 45 GBP. x =    km
The starting fare of MiniCabs is 2.00 GBP higher than of BlackCabs, however the ride fare is lower at 1.40 GBP per kilometre travelled.
4.Find the least number of whole kilometres required to travel for the total cost to be less on a MiniCab ride in comparison to a BlackCab ride.

  Two identical water waves pass a sensor in an oceanography laboratory. A 5mg4o4ib(p4bp hpv m2,hg7mb s the waves pass the sensor, the depth, D, in metres, of water is recorded and modellebp gbhb4,4 2 givpp(5o mm7hm4d by the function
D(t)=3.21−0.45cos($24.2^{\circ}$ $\times t$)where t is the elapsed time, in seconds, since the first wave hit the sensor.
1.Find the minimum and maximum depths of the water as the two waves pass the sensor. $D_{min}$ =    m $D_{max}$ =    m
2.Find the first time after 1616 seconds at which the depth of water reaches 3.5m. t ≈   

  The most popular mobile phone plans in 2019 frob v154gjc57o wd1*q*r t,utujir4 vlbm a major mobile service provider in Australia are summarised in the table bel1vru1o* w4 lvcj45j b,r*b7iuq 5tdgtow.

1.The total monthly cost, C, in AUD, of the Optimal Data Plan can be modelled as a piecewise linear function of the amount of data, d, in GB, used. Determine the model.
2.Find down the total monthly cost of the Optimal Data Plan if:
(1)8 GB of data is used;    AUD
(2)12 GB of data is used. C =    AUD
3.Determine how many GB of data is used when the Optimal Data Plan becomes more expensive than the Massive Data Plan.
Daniela is using the Optimal Data Plan and is thinking of changing it to the MassiveMassive Data Plan. It is the evening of December 7, 2019 and she has already used 3.5 GB of data so far. Assume that her data usage continues at the same constant rate during December and the next year.d =   
4.Estimate Daniela's data usage in December.    GB
5.Hence explain why Daniela should change her plan.
6.Find the amount of money Daniela will save each month in 2020 if she changes her Optimal Data Plan to the Massive Data Plan.    AUD

  The Chicago Wheel, the world's fi(au aci nz ugu6fucn*( 5b924frst Ferris wheel, was designed and built in 1893 by an American civil engineer, George W. naf*9g(cu2ub au45 6( zuifn cG. Ferris Jr.
The height, in metres, above the ground of a passenger on the Chicago Wheel after
t minutes can be modelled by the function h(t)=38cos($\frac{π}{10}$​(t−10))+42.
1.Find the maximum height reached by a passenger on the Chicago Wheel. h =    m
2.Find the height above the ground of a passenger 1212 minutes after the ride has started. h ≈    m
3.Find the time, in minutes, it takes for the Chicago Wheel to complete one rotationrotation. t =    min
4.Given that passengers only complete one rotation on the Chicago Wheel, calculate how long they are more than 65 metres above the ground. $t_{more\,than}\,65_m$ ≈    min

  Certain content on the internet can become viral when the fhoi* ,ml)ud 7e1sh3gcontent spreads exponentially. The spread of content published by i)mf e7s,*ho dhlg13ua specific content creator can be modelled by the function
N(t)=145$\times1.25^t$,
where N is the number of people reached by the content, and t is the number of hours since the content's publication.
1.Write down the number of people reached immediately after the content's publication. N =   
2.Calculate the number of hours it takes for the content to reach 100000 people. t =   
3.Calculate the number of people reached after 80 hours.
Approximately 5 billion people in the world are now connected to the internet. N =   
4.Explain why the model starts to become unrealistic after about 3 days.

  The subscription fee for an online project management softwar pmb0glt-n.i 1e by AB-Tech is $40$40 per month. If the customer buys for a whole year in advance, a discount of $130$130 is applied.
This can be modelled by the following function, P(n), which gives the total cost when paying annually for the subscription.
P(n)=40n−130,n≥12where n is the number months.
1.Find the total cost of buying a subscription for 2 years. P =    dollars
2.Find $P^{−1}$(1790). $P^{-1}$ =    months
The subscription price for a different online project management tool by YZ-tech is 35dollars per month, however customers can only purchase annually in advance, and there are no discounts. The total subscription cost of YZ-tech's software is less than AB-tech's software when n>k .
3.Find the minimum value of k. k =   

  Consider the quadratic funco3tzc u1f6z0slr;l; ction f(x)=a$x^2$+bx+c. The graph of y=f(x) is shown in the diagram below. The vertex of the graph has coordinates R(m,−9).
The graph intersects the x-axis at two points; P(−4,0) and Q(2,0).
1.Find the value of m. m =   
2.Find the values of a, b, and c. a =    b =    c =   
3.Write down the equation of the axis of symmetry of the graph. x =   

  The graph below showfumsq22q3 kyr yo)(n9s the amount of money M (in thousands of dollars), in the account of a contractor, where t is the time in mono uq3(kysm y 2nrf9q)2ths since he opened the account.

1.Write down one characteristic of the graph which suggests that a cubic function might be an appropriate model for the amount of money in the account.The equation of the model can be expressed as M(t)=$at^3$+$bt^2$+$ct$+$d$, where a, b, c and d∈R. It is given that the graph of the model passes through the following points.

2.(1)State the value of d.
(2)Using the values in the table, write down three equations in a, b, and c.
(3)By solving the system of equations from part (ii), find the values of a, b and c. a =    b =    c =   
If M has a negative value, the contractor is in debt.
3.Use the model from part (b) to find the number of months the contractor expects to be in debt. Give your answer to the nearest month.

  Tom has a model train set, which includes a tunnel that the vj:14)dtp*vp xu zjw+ model train must pass throuj*:w4uxv )pj+zv1pd tgh. The entrance of the tunnel is in the shape of the parabola h(x)=−0.6$x^2$+3.12x, where ℎh is the height of the tunnel in centimetres, and x is the horizontal distance from the bottom left corner of the tunnel entrance at (0,0), also in centimetres.

1.Find the equation of the axis of symmetry of the parabola that models the tunnel entrance.
Tom is considering purchasing a new carriage for his model train. The carriage is in the shape of a cuboid with a height of 3.5cm and a width of 2cm. x =   
2.Determine whether the carriage will fit through the tunnel entrance.

  The cross-section of a ship's hulln2/pq ifjgzd1 d47ze a1o/l q6 below the surface of the ocean can be modelled by a parabola. The depth of the boat's hull, d metres, is given by nz aj2 i7de4/ /6gzq1dpfoq1 ld(x)=0.5$x^2^−6.4x, where x is the horizontal distance from the left-hand side of the hull at the surface, also in metres, as shown in the following diagram. The x-axis represents the ocean surface.

1.Find the equation of the axis of symmetry of the parabola that models the hull of the ship.
The ship has a horizontal storage deck at a depth of 1010 metres, with a ceiling at surface level. The storage deck is used to transport containers in the shape of cuboids. x =   
2.Determine the maximum width of a container that the ship can transport on the storage deck, given that the container is 1010 metres high. Give your answer to the nearest metre.

  The amount of fuel ikdm o0uhovm)q,an . wnh0.( 7en a car's tank, V, in litres, can be modelled by a linear function, V=mx+c, where x iso.vd e.mhh )uqwnn7a0 km(o ,0 the distance driven measured in kilometres.
After driving 100 km, the amount of fuel in the tank is 33 litres.
1.Write down an equation that describes this information.
After driving 250 km, the amount of fuel in the tank is 22.5 litres.
2.Write down an equation that describes this second piece of information.
3.Calculate the amount of fuel in the tank after driving 50 km. V =    litres

  The dietary reference intake ( ei0 2l(fdldgc6k:y a(DRI, in grams), y, of protein for a sedentary male adult of mass x kg is determined by the: l (60g l2fackyddi(e equation y=mx+c, where m and c are constants.
For a male adult of mass 60 kg the DRI of protein is 6969 grams, and for a male adult of mass 90 kg the DRI of protein is 91.5 grams.
1.Determine the value of m and the value of c.
2.Find the DRI of protein for a male adult of mass 70 kg. y =    grams
3.Suggest a reason why the equation may not be appropriate for determining the DRI of protein for a male child of mass 40 kg.

  Mehmet organises a graduation party on a boat iy(7wablmx v, o+zy o93w6dnznsrh(3 9 n the Strait of Istanbul for his peers from college. The cost to rent a large boat for the evening is 5000 Turkish liras (TRY) and the cost to hire a DJ is 1500 TRY. The cost for food and drinks is estimated to be 125 ox39syd zbw+onhzl 3 6(a7vr nym 9w(,TRY per person.
1.nd an expression for the total cost, y TRY, of the evening in terms of the number of peers, x, attending the party. y = a+bx a =    b =   
2.nd the total cost of the evening if 40 peers attend the party. y =    TRY
3.ven that Mehmet decides to collect 250 TRY from each peer attending the party, find the least number of peers he has to invite to be able to cover the total cost of the evening. x =   

Two IT consulting firms have different pricing structures. TechMind-vqtv, igw38 4tm3fbd charges 115 dollars,3 td,vtvmg- qwi8bf43 plus an upfront fee of 250 dollars for booking an appointment. RiddleBreak doesn't charge for booking an appointment, but the rate is 60 dollars per hour. Let $C_1$​(t) and $C_2$​(t) be the costs, in dollars, of consulting t hours by TechMind and RiddleBreak respectively.
1.Write down expressions for the costs of consulting, $C_1$​ and $C_2$​, in terms of the number of hours, t, charged.
2.Find the least number of whole hours of consulting from TechMind for its servicesservices to be a cheaper option.

Two carpet cleaning companies have diffeb0v0o d; (c9tg hxyf.xi47,px pgq u5hrent pricing structures. CarpetShine charges 1.801.80 AUD per square metre, plus an upfront scheduling fee of 5050 AUD. CleanMaster has nobf9qy ix4.dt7xpx,p cg5(h0o;0gvh u scheduling a service fee, but the rate is 2.50 AUD per square metre. Let $C_1$​(x) and $C_2$​(x) be the costs, in AUD, of cleaning x square metres of carpet area by CarpetShine and CleanMaster respectively.
1.Write down expressions for the costs, $C_1$​ and $C_2$​, of cleaning x square metres of carpet area.
2.Find the least number of whole square metres of carpet area cleaned by CarpetShineCarpetShine for its servicesservices to be a cheaper option.

  A cannon-ball is fired from the top of a tower. The heighy3rx.9hzp/oq t7 .p qy3kp.fct, ℎh, in metres, of the cannon-ball above the ground is modelled by the funct . pqorztcx y3h7pp9.y.k3/fqion
h(t)=−2$t^2$+20t+8,t≥0,​where t is the time, in seconds, since the moment the cannon-ball was fired.
1.Write down the height of the tower. h =    m
2.Calculate the height of the cannon-ball 5 seconds after it was fired.
The cannon-ball hits its target on the ground n seconds after it was fired. h =    m
3.Find the value of n. n ≈   
4.Find h′(t). h′(t) = at +b a =    b =   
5.Calculate the maximum height reached by the cannon-ball, and write down the corresponding time t.
6.Determine the total time the cannon ball was above the height of the tower. t =    seconds

  The graph of a quadratic function has a y-intercept at Ay/ae tc lui2up,aq9m; zgpl-; 4+gd b+(0,24) and one of its x-intercepts is B(2,0). t+9u;ym-ugi +dla a ;2l,pgp q4e bz/c
The x-coordinate of the vertex of the graph is 4.
The equation of the quadratic function is in the form y=a$x^2$+bx+c.
1.Write down the value of c. c =   
2.Find the value of a and the value of b. b =   
3.Write down the coordinates of the second x-intercept of the function.
(a,b) a =    b =   

  The number of French words, N, that are remembered by stuvync7bx /f.*z)c6kk adents after the completion of a French language course decreases exponentially over time. This data can be modelled by the funy )vfncb 6xk /.*zack7ction
N(t)=a$\times$b$^{-t}$+450,​where a and b are positive constants, and t is the time in years since a student completed the French language course.
Immediately after completion, a student remembers 4200 French words.
1.Find the value of a. a =   
After 4 years a student remembers only 1600 French words.
2.Find the value of b. b ≈   
The number of French words a student remembers never decreases below a certain number of words, n.
3.Write down the value of n. n =   

  Sarah's investment portfolio, V(t), in Australian dollars (AUD), is model qrb/j.l isj83led by si /jjq8 3b.rlthe function
V(t)=64000$\times$(1+$\frac{k}{1000}$)$^t$,t≥0,where t is the number of years since 2019 and k is a constant.
1.Find an expression for the value of the investment portfolio in 2020. Give your answer in the form a+bk, where a,b∈$Z^+$.
Three years later, in 20222022, the value of the investment portfolio is expected to be 74088 AUD.
2.Find the value of k. k =   
This model is defined for 0≤t≤n. At n years, the value of the investment portfolio would have doubled since 20192019.
3.Find the value of n. n ≈    years

  The water depth, D, in metres, in a boating marina on a particular day can be q cmx;soq r,.)yb +:0snq:m;m v4gxcv model4vbnmmoc.gq 0q:: xyq;vs,rms)c; x+ led by the function
D(t)​=4sin($30^{\circ}$$\times$t)+12.5,0≤t≤24,​where t is the elapsed time, in hours, since midnight.
1.Write down the depth of water at midnight.   
2.The cycle of water depths repeats every T hours. Find the value of T. T =   
3.(1)Write down the minimum and maximum depths of water during the day. $D_{min}$ =    $D_{max}$ =   
(2)Find the times at which the minimum and maximum depths of water occur during the day.
4.Draw the graph of D versus t on the grid below.

5.A ship with 8 m draught is allowed to enter the marina if the depth of water is greater than 1010 m. Find the time interval before midday, in hours and minutes, during which the ship is not allowed to enter the marina.

  The water depth, D, in metres, j2pb 6ed+tpz6 q8x w2mat the entrance of a boating harbour can be modelled by the function2dqt bmp +xe8w2jp66z
D(t)​=4.5cos($30^{\circ}$$\times$t+$90^{\circ}$)+15,0≤t≤24,​where t is the elapsed time, in hours, since midnight.
1.Write down the depth of water at midnight.   
2.The cycle of water depths repeats every T hours. Find the value of T. T =   
3.(1)Write down the minimum and maximum depths of water during the day. $D_{min}$ =    $D_{max}$ =   
(2)Find the times at which the minimum and maximum depths of water occur during the day.
4.Draw the graph of D versus t on the grid below.

5.A ship with 10 m draught is allowed to enter the harbour if the depth of waterwater is greater than 12 m. Find the time interval after midday, in hours and minutes, during which the ship is not allowed to enter the harbour.

  The bench press world record weight, W, in kilograms, 369od3xkk umzis modelled by the function om kux3 d9z63k
W(t)=$\frac{M}{1+Ce^{-0.04t}}$,
where t is the number of years that have passed since 1953.
The first bench press world record of 227.27 kg was set in 1953. According to bench press experts, the record weight will eventually reach a limit of 350 kg.
1.(1)Write down the value of M. M =   
(2)Find the value of C , giving your answer correct to 2 decimal places. C ≈   
2.The current unbeaten bench press world record weight is 337 kg. Find the year in which this record was set.
3.A young and promising powerlifter claims that he is going to break any bench press world record performed until 2025. Find the weight that he has to bench press in 2025 to set a new bench press world record according to the model. W =   

  An elephant raised in a wil yt+.dx7thz(hdxa b )/dlife sanctuary. Her height, H, in metres, is modelled by the logistic funct(z7 d.)/hahtx dyxtb+ ion
H(t)​=$\frac{2.5}{1+1.5e^{-0.25t}}$​,
​where t is the number of years that have passed since her birth.
1.Find the elephant's height:
(1)when she was born; H =    m
(2)on her 55th birthday. H ≈    m
The elephant's height approaches a limit of M metres as she ages.
2.Write down the value of M. M =   
The elephant reaches a height of 22 m before her $n$th birthday.
3.Find the smallest value of n. n =   

  The mass of a radioac )xwjnvdez .+9tive sample is given by
M(t)=P+$Q(\frac{1}{2}​)^{\frac{t}{300}}$​,t≥0 days.where t is the number of days after the sample was collected and measured.
The initial weight of the sample is 20 g. When each atom emits its α-particle, the mass of the atom is reduced by 3.2%.
1.(1)Explain why P is the final mass of the sample after every atom in the sample has emitted its α-particle.
(2)Hence, find the value of P, leaving your answer correct to two decimal places. P ≈    g
2.Find the value of Q. Q =   
3.Estimate the mass of the sample after 2.5 years (assume no leap years). M ≈    g

  An oil tank at a mine site was at full capacity before the tal:a3n8 wx .c8u0cvq+ thcbj8znk incurred a puncture in the base and the oil start to leak out. A site engineer used the following function, L(t), to model the percentage of oil remaining in the ta 3 jz :hwtn8caqu+ v0cxb.lc88nk
L(t)=$100e^{−kt}$,t≥0,where t is the number of days after the puncture occurred.
The engineer found that after one day, 30% of the oil originally in the tank had leaked out.
1.Find the value of k. k ≈   
2.Use this model to find the percentage of oil remaining in the tank after 30 hours.
Based on the model, the engineer makes the claim that the tank will always have some oil in it and never completely empty. L =   %
3.State a mathematical reason supporting the engineers claim.

  The following table shows the total revenue btop.j b9ko6n u7h+ 1rvi:xw(, y, in Australian dollars (AUD), obtainedobtained daily during the first week of January 2020, by Peppy's Pizza restaurantrestaurant and the number on+ b:roj vhk i7xp6tu91 (wo.bf guests, x, served.

1.(1)Calculate the Pearson's product-moment correlation coefficient, r, for this data. r =   
(2)Hence comment on the result.
2.Write down the equation of the regression line y on x. y =ax+b a =    b =   
3.Use the line of the regression to estimate the revenue of serving
70 guests. Give your answer correct to the nearest AUD.
Daily maintenance cost for the restaurant is 240 AUD. Additionally, the cost of serving one guest is 5 AUD. y ≈   
4.Determine if the restaurant makes a profit when serving 45 guests on a particularparticular day.
5.(1)Write down an expression for the total revenue of serving x guests.ax+b a =    b =   
(2)Find an expression for the profit of the restaurant when serving x guests on a particular day.ax+b a =    b =   
(3)Find the least number of guests required to be served to result in a profit for the day. x ≈   

  The following table shows the total reven2/tqgsdx2pwd1-s; w/(w gv ugue, y, in US dollars (USD), obtainedobtained monthly during the first six months of 2020, by Law Office of Fox Brothsv- xtdd/wg;1( 2 s/qgw2wugp ers and the number of clients, x, served.

1.(1)Calculate the Pearson's product-moment correlation coefficient, r, for this data. r =   
(2)Hence comment on the result.
2.Write down the equation of the regression line y on x. y =   
3.Use the line of the regression to estimate the revenue of serving
20 clients. Give your answer correct to the nearest USD.
Monthly operating cost for the law office is 2500 dollars. Additionally, the cost of serving each client is 200 dollars. y =ax+b a =    b =   
4.Determine if the law office makes a profit when serving 66 clients on a particularparticular month.
5.(1)Write down an expression for the total revenue of serving x clients. y =ax+b a =    b =   
(2)Find an expression for the profit of the law office when serving x clients on a particular month. y =ax+b a =    b =   
(3)Find the least number of clients required to be served to result in a profit for the month.

  The data below shows the price, y, in Kazakhxd ded2jw-.8 rstani tenge (KZT), that each bag of wdedr 2wj x8.-dheat flour can be purchased for at a wholesale market in Almaty if x bags are ordered.

1.Find an expression for the price, y KZT, of each bag purchased in terms of the number of bags, x, ordered. y =ax+b a =    b =   
2.Hence find the exact total cost of purchasing 250250 bags of wheat flour. Total cost =    KZT

  Part of the graph of y=f(x nktc k*vjh6)ubx;1/l) is shown below. The point A(3,4) lies on the graph. uc/* h ktbl1kxvj6;n)

1.Write down the value of f(3). f(3) =   
The tangent line L to the graph of y=f(x) at point A has equation y=3x−5.
2.Find the equation of the normal line N to the graph of y=f(x) at point A. Give your answer in the form ax+by+d=0, where a,b,d∈Z. ax+by+c=0 a =    b =    c =   
3.Draw the lines L and N on the same grid above, labelling the intercepts with the axes.

  The number of users on a2 gcprq4+ax0s new social network in 2018 was 320000. One year later, in 2019, the number of users on this network is estimated to be 350000. The numbera 40sqcrxg2p+ of the users on this network, N, can be modelled by the function
N(t)=320000$\times$$b^t$,t≥0,​where t is the number of years since 2018 and b is a constant.
1.Find the value of b. b ≈   
2.Estimate the number of the users on this network there will be in 2023. N ≈    users
3.Forecast the year for which the number of the social network users reaches one million. Give your answer correct to the nearest year.

  The downward speed, V, in metres per second, gaine +z+rv34zbfrfd by a skydiver during a jump is mode vfb3f++4r zzrlled by the function
V(t)=56−56$\times3$$^{-\frac{t}{4}}$,t≥0,​where t, in seconds, is the time for which the skydiver is free falling in the air.





1.Write down the downward speed of the skydiver at t=0.
The line L is the horizontal asymptote to the graph. V =   
2.Determine the equation of L.
The skydiver starts to feel no acceleration when he reaches the speed of 55 metres per second. y =   
3.Find the time it takes for the skydiver to reach this speed. Give your answer correct to the nearest second. t ≈   

  The average petrol price, P, in cents per litre, in Australia 2019 cas2v+duu.k k*in be modelled by th+du.ksk2 i u*ve function
P(t)=asin($10^{\circ}$$\times$t)+b,0≤t≤365,where t is the elapsed time, in days, since New Year's Eve 2018.
Part of the graph of y=P(t) is shown below.




1.Write down the value of b.
The average price of petrol on the 21st of January, 2019 was 139 cents per litre.   
2.Show the value of a is equal to 12. a =   
3.Hence calculate:
(1)the average petrol price on the 10th of February, 2019; P ≈    cents per gallon
(2)the minimum and maximum average prices of petrol in Australia 2019.
The cycle of average petrol prices repeats every T days. $P_{min}$ =    cents per gallon $P_{max}$ =    cents per gallon
4.Determine the value of T. T =   

  The average gasoline price, P, in cents per gallon, in USAnfnimbb.t-8mdvl0/ - 2019 can be modelled by the func/0m.bfn-mbil - t8vn dtion
P(t)=acos($15^{\circ}\times$t+$90^{\circ}$)+b,0≤t≤365,
where t is the elapsed time, in days, since New Year's Eve 2018.
Part of the graph of y=P(t) is shown below.

1.Write down the value of b.
The average price of gasoline on the 14th of January, 2019 was 328 cents per gallon.   
2.Show that the value of a is equal to 56. a =   
3.Hence calculate:
(1)the average gasoline price on the 1414th of February, 2019; P =    cents per gallon
(2)the minimum and maximum average prices of gasoline in USA 2019. $P_{min}$ =    cents per gallon $P_{max}$ =    cents per gallon
The cycle of average gasoline prices repeats every T days.
4.Determine the value of T. T =   

  The table below shows the income tax rates in Aus7,vuly44( y)jad ayfvtralia for 2019/2020.

1.Find the exact amount of tax payable for a person who earns 200000 dollars.    dollars
2.Draw the graph of tax payable versus income on the grid below.

  When a ball is thrown frpl rt9i/c.nmw 9-jq,ta pl.g/om the top of a tall building, its height above the ground after9l/ pttm, n /rajil gwp9.-.qc t seconds is given by
s(t)=$at^2$+bt+c, where a,b,c∈R and s(t) is measured in metres. After 1 second, the ball is 84.3 m above the ground; after 2 seconds, 93.9 m; after 8 seconds, 42.3 m.
1.(1)Write down a system of three linear equations in terms of a, b and c.
(2)Hence find the values of a, b and c. a =    b =    c =   
2.Find the height of the building. s =    m
3.Find the time it takes for the ball to hit the ground. t ≈    seconds

  When a coin is thrown from the top of a skybgty7 .frg1flh8dxs 69o6qf 1 scraper, its height above the ground after t seconds is given fry7g6 qs9l. f16d8ghof1 txbby s(t)=$at^2$+bt+c, where a,b,c∈R and s(t) is measured in metres. After 11 second, the coin is 179.3 m above the ground; after 2 seconds, 188.2 m; after 66 seconds, 159.8 m.
1.(1)Write down a system of three linear equations in terms of a, b and c.
(2)Hence find the values of a, b and c. a =    b =    c =   
2.Find the height of the skyscraper. s =    m
3.Find the time it takes for the coin to hit the ground. t ≈   

  A farmer is going to fence two equal adjacent parcels of land. These parcelsg d5jw h+6iufom,:mp9 share one side (which also requires fencing) as shown in the following diagram. The farmer has only 80 me95jwp u: ifh+omg 6m,dtres of fence.

1.Write down the equation for the total length of the fence, 80m, in terms of x and y. ax+by=c a =    b =    c =   
2.Write down the total area of both parcels of land in terms of x.
3.Find the maximum area, in $m^2$, of one parcel of land. ≈   

  The axis of symmetry of the graph of a quadratic function has equa r.m,v(adn: ndtion x=$\frac{3}{2}$​.
1.Draw the axis of symmetry on the grid below.

The graph of the quadratic function intersects the x-axis at the point P(−1,0). There is a second point, Q , at which the graph of the quadratic function intersects the x-axis.
2.Mark and label P and Q on the grid above.
The graph of the quadratic function has equation y=$−x^2$+bx+c, where b,c∈Z.
3.(1)Find the value of b and the value of c. b =    c =   
(2)Find the coordinates of the vertex, M. (a,b) a=    b=   
(3)Draw the graph of the quadratic function on the grid above.

  The graph shows the curve of a quadratim l o-tb)9m:4mx/wlib c function of the form f(x)=$ax^2$+bx+90.

1.Write down the equation for the axis of symmetry of the curve. x =   
2.Hence, or otherwise, find the value of a and the value of b. a =    b =   
3.Find the y-coordinate of the vertex of the curve. f(x) =   

  A population of goldfish decreases exponentially. The size of the population, P,,tu4gl8gw: / :l7l- oyej udqx after t days is modxl7guy lt :dul:q,woe - j48g/elled by the function
P(t)=8000$\times2^{-t}$+100,t≥0.​1.Write down the exact size of the initial population. P =   
2.Find the size of the population after 5 days. P =   
3.Calculate the time it will take for the size of the population to decrease to 120.
The population will stabilize when it reaches a size of n. t =    days
4.Write down the value of n. n =   

  The half-life, T, in years, of a radioactive isotope can be modelled by v;u9h2ge 1k b+d6ohxy the functv6kdbhg y h92xo; 1+euion
T(k)​=​$\frac{ln0.5}{ln(1-\frac{k}{100})}$,0where k is the decay rate, in percent, per year of the isotope.
1.The decay rate of Hydrogen-33 is 5.5 % per year. Find its half-life.
The half-life of Uranium-232 (U-232) is 68.9 years. A sample containing 250 grams of U-232232 is obtained and stored as a side product of a nuclear fuel cycle. ≈   
2Find the decay rate per year of U-232. k =    %
3.Find the amount of U-232 left in the sample after:
(1)68.9 years;    grams
(2)100 years. ≈    grams

  The half-life, T, in days, of a radioactiv.hlw5)uyapy8 rxfz,y4+ 9sle :lxrgc5i -f(f e isotope can be modelled by the function ulg4h5 ( ,e-pxz:ff cfxyr9i wlya) +sy 5l8.r
T(k)​=​,01.The decay rate of Gold-196 is 6.2 % per day. Find its half-life.
The half-life of Phosphorus-32 (P-32) is 14.3 days. A sample containing 120 grams of P-32 is produced and stored in a biochemistry laboratory. T ≈   
2.Find the decay rate per day of P-32. k =    %
3.Find the amount of P-32 left in the sample after:
(1)14.3 days;    grams
(2)50 days. ≈    grams

  Radiocarbon dating can be used to estimate the age of an)4kfg m)px(37sof1cycfkk8 w artifact made of wood. Archeologists take a sample of wood from the artifact and measure the ratio,r, of Carbon-1414 in the sample to Carbon-1414 in a living woodpxf)y 34gcf(kc s1w7omk8) fk. The age, t, in years, is then estimated by using the equation
t​=−​$\frac{6000}{ln2}\times$lnr,[Age Model]
​where 0.01A team of underwater archeologists found an ancient wooden shipwreck deep down in the Black Sea. A sample of wood is taken from the shipwreck to determine its age.
1.Find the age of the shipwreck if r=0.75.
In all wood samples, Carbon-14 decays over time. If r<0.01, there is insufficient amount of Carbon-14 left in the sample wood for its age estimate calculated by the age model to be reliable. t ≈   
2.Find the oldest sample of wood for its age estimate found by the age model would just cease to be reliable. Give your answer to the nearest10000 years. t ≈   

3.Using the age model, express r in terms of t.
4.Hence find r when t=7500, giving your answer correct to 2 decimal places. r ≈   

  The frequency, f, in her minkwzpfkg:88 j)9n;tz (Hz), and the period, T, in seconds, of a wave are related by the eq 9)kz8 w;jnkm8np:igf uation
f=$\frac{1​}{T}$An electronic signal is sent out with amplitude3 and frequency 200 Hz. The strength of this signal at time t seconds is given by $S_1$​=3sin(bt)
1.Find the value of b, giving your answer in terms of $π$.    $π$
A second signal of equal strength is sent out with a time delay of 2 milliseconds. The strength of the second signal at time t seconds is given by
$S_2$​=3sin(b(t−c))
2.Write down the value of c.    seconds
The strength of the sum of the two signals at time t seconds is given by
$S_3$​=3sin(bt)+3sin(b(t−c))
3.Draw the graphs of $S_1$​, $S_2$​ and $S_3$​ versus t, for 0≤t≤0.01, on the same axes.

  ­A population of 40 rabbits are brought to an island at the enpz:;k luxj)n50axssl2,c5 i xd of December, 2018:,x c;l 5p ax2x0sz5i ks)lunj.
The population, P, of rabbits on the island can be modelled by the logistic function
P(t)​$\frac{M}{1+Ce^{-0.3t}}$=​,​wheret is the number of months that have passed since New Year Eve 2019.
After 12 months, the population of rabbits on the island is increased to 1423.
1.Find the value of M and the value of C. M ≈   
2.Write down the largest population of rabbits that the island will ever be able to accommodate.   
3.Hence calculate the number of months it takes for the population of rabbits to reach half of its largest population. t =    months

  The position,s, in metres, of a cyclist ozpsrfg kbt9oldec8vp0) 24 .0 n a straight track can be modelled by the piecewise functionwhere t is the ti4v d8tp9klgbf .pe02ocz)0r sme, in seconds, and a,m∈R.





1.Show that a=0.1995 correct to 44 significant figures.
2.Find the exact value of m. m =   
3.Calculate the time it takes for the cyclist to travel: s =   
(1)15 m; t ≈    seconds
(2)200 m.t ≈    seconds
4.Draw the graph of s versus t, showing the position of the cyclist during the first 20 seconds.
5.Determine the time interval when the cyclist is travelling fastest.
6.Hence find its maximum speed, giving your answer in kilometres per hour.    km$h^{-1}$

  A healthcare company is designing a new d,c(6.(l xho gaccy 0gzeodorant stick that can be modelled by the shape of a parabola as the top, vertical lines as the sides and a horizontal line as the base, on the x-axis. The parabola has end points at (0,7) a0c(agzhc logcx,y(.6nd (5,7), and vertex at (2.5,8).
This design is shown in the diagram below. All distances are measured in centimetres.



The quadratic curve can be expressed in the form y=ax2+bx+c for 50≤x≤5.
1.(1)Find the value of c. c =   
(2)Using the points(2.5,8) and (5,7), write two equations in a and b.
(3)Hence, find the equation of the quadratic curve. y=$ax^b+cx+d$ a =    b =    c =    d =   
2.Find the area of the shaded region in the deodorant design. Area ≈    cm$^2$

  A factory produces bars in the shape of a cuboid,rl m-4:a1dnn;e . jdoq:qt6 bd with a fixed height of 30 cm and varying base. The area, A, of each bar b .qj6m- n;t:dre4q1bolda:nd ase is described by the function
A(x)​=x(40−x),5≤x≤35,​where x is the length of the bar base in centimetres.
1.Bar P has a length of 10 cm. Find the volume of P.    $cm^3$
2.Bar Q has the same volume as bar P, however has a different length. Find the length of Q.    cm
3.Find the value of x that makes the bar's volume a maximum, and state this maximum volume.

  A right cylinder has height ℎh mm ands0mdbn y9+ nhk*g :7pm diameter x mm. The volume of this cylinder is equal +gby *snh 9:p md7m0knto 45 $mm^3$.


The total surface area, A, of the cylinder can be expressed as A=$\frac{π}{2}​x^2$+$\frac{k}{x}$​.
1.Find the value of k. k =   
2.Find the value of x that makes the total surface area a minimum.

  A loaf pan is made in the shape of a cylinder. The pan has a diameter of 2: k0:1y bsvt zgv996jkhde-qk4 cm and a heigkjd61 :k-v 9z g9sh:0vbyetkqht of 5 m.
1.Calculate the volume of this pan.
Gloria prepares enough bread dough to exactly fill the pan. The dough was in the shape of a sphere. $V_{pan}$ ≈   
2.Find the radius of the sphere in cm, correct to one decimal place.
The bread was cooked in a hot oven. Once taken out of the oven, the bread was left in the kitchen.
The temperature, T, of the bread, in degrees Celsius,$^{\circ}C$, can be modelled by the function
T(t)=$a\times(1.51)^{−\frac{t}{3}}​$+21,t≥0,
where a is a constant and t is the time, in minutes, since the bread was taken out of the oven.
When the bread was taken out of the oven its temperature was $205^{\circ}C$ r ≈    cm
3.Find the value of a. a =   
4.Find the temperature that the bread will be 10 minutes after it is taken out of the oven.
The bread can be eaten once its temperature drops to $35^{\circ}C$.T ≈    $^{\circ}C$
5.Calculate, to the nearest minute, the time since the bread was taken out
of the oven until it can be eaten. t ≈    minutes
6.In the context of this model, state what the value of 21 represents.

  A hot apple pie was taken out of the oven and left cooling on the bench. The tembu*yck9,cq ,s peraturetb,c*q,cysu9 kemperature of the kitchen is $19^{\circ}C$ This situation can be modelled by the exponential function T=a+b($k^{−t}$), where T is the temperature of the appleapple pie, in$^{\circ}C$, and t is the number of minutes for which the apple pie has been on the bench in the kitchen. A sketch of the situation is given below.

1.Explain why a=19.
Initially, at t=0, the temperature of the apple pie is $180^{\circ}C$.
2.Find the value of b.
After being left cooling on the bench for one minute, the temperature of the apple pie is $159^{\circ}C$. b =   
3.Show that k=1.15.
4.Find the temperature of the apple pie five minutes after it has been left cooling on the bench. T ≈    $^{\circ}C$
5.Find the total time needed for the apple pie to reach a temperature of $30^{\circ}C$. Give your answer in minutes and seconds, correct to the nearest second. t ≈    min

  The temperature, T, in degrees Celsius4 u k*,aa(mapsh4 ;esz, in Istanbul during a particular winter day is modelled by th a;4ks4as,mua p*z(ehe equation
T​=asin(b(t−c))+d,0≤t≤24,​where a,b,c,d∈R and t is the elapsed time, in hours, since midnight.
The graph of T versus t is shown below.

The lowest temperature was recorded at 33:3030 am and the highest temperaturetemperature was recorded at 33:3030 pm.
1.Write down the value of:
(1)a;   
(2)d.   
2.Find the value of b. ≈   
3.Find the smallest possible value of c, given c>0.
The temperature, T, in degrees Celsius, in Istanbul during a particular summer day is modelled by the equation
T​=5sin(0.262(t−8))+25,0≤t≤24,
​where t is the elapsed time, in hours, since midnight. c =   
4.Find the time, in hours and minutes, when the temperature:
(1)reaches its maximum; ≈    pm
(2)first drops below $244^{\circ}C$. ≈    pm
5.The temperature is below $24^{\circ}C$ for h hours and m minutes. Find
the value of h and the value of m. h =    m =   

  The tides at the Port of Bristol, UK, w :bwrrt2is:/lazyu9 ( ere observed by a student.
On a particular day, the range between the lowest and highest tides is 10.6 m and the time difference between high tides is 12.2 hours. The first highest tide occurs at 8:12 am and is 12.5 m high.
The height, H, in metres, of the tides can be modelled by the equation
H​=asin(b(t−c))+d,0≤t≤24,​where t is the elapsed time, in hours, since midnight.
1.Write down the value of:
(1)a;   
(2)d.   
2.Find the value of b. ≈   
3.Find the smallest possible value of c, given c>0. c ≈   
4.Hence draw the graph of H versus t, for 0≤t≤24.

  The downward speed, V, in metres per second, of a skydiver t seco3kd8uyw wnyy:)4dqn;nds after a jump from wdd y;)nky8n34q :uwy a plane is modelled by the piecewise function


1.Show that K=60.
2.Draw the graphs of y=60−$\frac{60}{t+1}$​, for t≥0, and y=12+$\frac{46}{t-28}$​, for t>28,
on the grid below.



3.Find the time T at which the skydiver opened their parachute.
4.Calculate the speed of the skydiver after:
(1)10 seconds; V ≈    m sec$^{-1}$
(2)32 seconds. V ≈    m sec$^{-1}$
Assume that the skydiver's parachute failed to open.
5.Write down the terminal downward speed of the skydiver.    m sec$^{-1}$

  The Burj Khalifa, located in Dubai,qfd4 vsadq )e6;la;o- is the tallest building in the world. It has a height of 830 m and has a square base that covers a; v-qqaodle6df;a 4)s floor area of 556 m$\times$556 m. A tourism shop located near the building sells souvenirs of the tower, which sit inside glass pyramids, as illustrated by the diagram below. The souvenir tower is an accurate scale replica of the actual tower.



The scaled model of Burj Khalifa has a base area of 20 cm$\times$20 cm. The height and base area dimensions of the glass pyramid are 10% larger than the model.
1.(1)Find the height of the souvenir tower, in cm, correct to the nearest mm. $h_m$ =    cm
(2)Find the volume of the glass pyramid, rounding your answer correct to the nearest cubic centimetre. V ≈    $cm^3$
The shop owner aims to maximise profits from selling the souvenirs. The more the owner orders from the manufacturer, the cheaper the souvenirs are to buy. However, if too many are ordered, profits may decrease due to surplus stock unsold.
The number of souvenirs ordered from previous years and the resulting profits are shown in the following table.

Quantity is 5000 ; Profit is 35000
Quantity is 10000 ; Profit is 95500
Quantity is 13000 ; Profit is 116500

The shop owner decides to fit a cubic model of the form
P(x)=$ax^3+bx^2+cx+d$ to model the profit, P, for x thousand souvenirs ordered.
2.Explain why d=0.
3.Construct three linear equations to solve for a, b and c, and hence write down the completed function P(x).
4.Find $P′(x)$.
5.Find, to the nearest hundred souvenirs, the optimal number of souvenirs the owner should buy to maximise profit, and the resulting profit from this number.    dollars

  In a water filtration plant, the containers of water are filled and then filtereho -xn1(zjp. fk zg5)td, all ktp j-h.xz f)zg n(15oby an automated system. The volume of water in a container, W(t), in litres, after t seconds of the filling commencing can be modelled by the function
W(t)=V(1−$e^{\frac{-t}{k}}$​),where V is the maximum volume of the water container and k is a positive constant used to alter the rate at which water is filling the containers.
The value of k is currently set at 25.
1.Find how long it takes, t, for a container to fill up to 50% of its maximum volume. t ≈    seconds
2.If one of the containers has a maximum volume of 20 litres, find the volume of water in this container after 22 seconds. W ≈    litres
The engineers of the plant want to alter the value of k so that it takes 20 seconds to fill each container up to 95% of their maximum volumes.
3.Find the value of k required. k ≈   

  A particular species ofz oq26.sd acac7v1 /jbc3 -eue microbes is studied in the laboratory. Microbes are grown under controlled conditions in pure culture. The number of microbes, N, present in the culture t hours after the start of the expero 162-ezaqbsd.jcc/a7vu ec3 iment are recorded. The results are shown in the table below.




1.Complete the third row of the table above.
2.Draw a scatter diagram of logN versus t, scaling the axes if needed.
3.State the type of model that best fits the data displayed on your scatter
diagram from part (b).
A scientist in the laboratory claims that the number of microbes, N, grown in the laboratory during the experiment can be modelled by an exponential function of the form N(t)=a$\times$$b^t$, where a and b are positive constants.
4.Explain why the scientist is correct.
5.Find the value of a and the value of b. a ≈    b ≈   

  A new class of antibiotics is studied in the medical re b 4f5 zt0yk83c.muextg.r e(e5)p bimsearch center. Antibiotics are applied to a pure culture of bacteria. The number of bacteria, N, present in the culture t hours after the startkrzm (t. buxi0eept5)ge 83 mbf4y5c. of the experiment are recorded. The results are shown in the table below.



1.Complete the third row of the table above.
2.Draw a scatter diagram of logN versus t, scaling the axes if needed.
3.State the type of model that best fits the data displayed on your scatter
diagram from part (b).
A researcher in the center claims that the number of bacteria, N, left in the culture afterafter applying antibiotics can be modelled by an exponential function of the form N(t)=a×bt, where a and b are positive constants.
4.Explain why the researcher is correct.
5.Find the value of a and the value of b. a ≈    b ≈   

  The amount of water, A, in thousands of litres, available inn+r 64 bz w0 )axzb jm9x*ohzfxj0yx5: a water tank located on a farm flucfz+49xrb h) yxxzj j0a5 nb0o:wzx6*m tuates in a yearly cycle and can be modelled by the function
A(t)=asin(kt)+b,where t is the elapsed time, in weeks, since the start of the year.
The amount of water available in the tank on week 6 is 24 thousand litres and on week 31 is 9.2 thousand litres.
1.Find the value of k, in degrees, assuming there are 52 whole weeks in a year. k ≈    $^{\circ}$
2.Set up a pair of equations to find the value of a and the value of b. Give your answers correct to the nearest integer. a ≈    b ≈   
3.Hence find the amount of water available in the tank in week 42. A =    thousand litres
4.Draw the graph of y=A(t) on the grid below, for one full year, indicating clearly the minimum and maximum points.

  The mean depth, D, in metres, of a mounts 62nt * lxzj.wd(w*ytain lake fluctuates in a yearly cycle and can be modelledx2tw6jd.*zt*wsy(l n by the function
D(t)=acos(kt)+b,where t is the elapsed time, in months, since the beginning of an autumn season.
The mean depth of the lake on month 1 is 33.2 m and on month 5 is 22.8 m.
1.Find the value of k, in degrees. k =    $^{\circ}$
2.Set up a pair of equations and find the value of a and the value of b. Give your answers correct to the nearest integer. a ≈    b ≈   
3.Hence find the mean depth of the lake on month 8. D =   
4.Draw the graph of y=D(t) on the grid below, for one full year, indicating clearly the minimum and maximum points.

The security personnel at the entry of a rural campground activate a +.2/x*d2rovwbpf/0 owt( :pm al fi wcsiren to alert campers when a wild animal is spotted nearby. The sound intensity, I, of the siren varies inversely with the square of the distance from the siren, d. When initially testing the siren, the security personnel found that at a distance of 3 metres from the siren, the sound intensity is /i l t f*w2 pr.:mwoa0pb+wdo/2cfx(v 5 watts per square metre ($Wm^{−2}$).
1.Show that I=$\frac{45}{d^2}$​.
2.Sketch the curve of I for d>0, labelling the point (3,5).
The campers can only hear the siren if the sound intensity at their location is greater than $1.2\times10^{−5}Wm^{−2}$.
3.Find the minimum distance, in kilometers, from the siren where the campers can no longer hear the siren.

The distance travelled, d, in meters, of a freely falling objecida 0f;4v5- wunh;)zjp oqjll9lsz+ 6t is proportional to the square of tl9wvq4ljo+n uf;;p a)id j65zhl0z- she time of the fall, t, in seconds.
A football dropped from the top of a school building falls a distance of 19.6 m in the first 2 seconds.
1.Show that d(t)=4.9$t^2$.
2.Sketch the curve of d for t>0, clearly showing the point (2,19.6).
The distance covered, s, in meters, by a quad-copter drone undertaking a vertical landing can be modelled by s(t)=8t, where t is the time in seconds after the landing commences.
The football is dropped at the same time as the drone commences a landing from the same height of the school building.
The football covers more distance than the drone when t>p.
3.Find the value of p.

  Peter wants to reach a rock, located 55 km away from his current positicoe.hxp;p wz9eh9,q6 on, using the stratew ; 9eehzp. h,opcq96xgy outlined below.
Every hour Peter approaches the rock by a fixed percent of the remaining distance to the rock.
The distance, D, in kilometres, Peter has travelled from his original position, can be modelled by the exponential function
D(t)=a+b($k^{-t}$),t≥0,​where t is the time, in hours, since Peter started moving.
1.Explain why a=5.
2.Find the value of b. b =   
After one hour, the distance from Peter to the rock is 3.75 km.
3.Find the value of k. k =   
4.Find the distance between Peter and the rock after 4 hours.
Peter can be heard when shouting from 200 metres away.    km
5.Calculate the time it takes until someone standing on the rock can hear Peter shouting. Round your answer correct to the nearest hour.

  The sound intensity level (SIL), L, in decib3o t jsouz .fa7gsa: *1v t*ctnq45au5els (dB), of the sound intensity, I, in watts per stj5*gc atfa z:7n o4vs1ta .*5 3uquosquare meter, is given by the equation
L​=10log($\frac{I}{I_0}$​),​where $I_0$​ is the reference sound intensity.
1.Two sounds have intensities $I_1$​ and $I_2$​ with $I_1$​<$I_2$​. Show that the difference
in SIL between the two sounds is L=10log($\frac{I_2}{I_1}$​).
2.A train rumble has the SIL of 82 dB and a dishwasher noise has the SIL of 52 dB. Find the number of times the train rumble is more intense than the dishwasher noise.
Jack is attending a rock concert where the SIL of the music reaches 110 dB. During the concert, Jack's mom calls him to find out if he is safe and everything is going well. The SIL of his cell phone ring is 70% of the SIL of the rock music. $I_2$ =    $I_1$
3.The perceived loudness of a sound doubles for every 10 dB increase in SIL.
(1)Write down the SIL of Jack's cell phone ring. L =    dB
(2)Jack perceives the rock music k times as loud as his cell phone ringing. Find the value of k, giving your answer correct to 3 significant figures. K ≈   
(3)Hence, decide whether Jack's mom will reach him during the concert if Jack carries his cell phone in the pocket and turned vibrating alert off.

  Alexandros monitors the temperature during a part,ngeq2rni l9 snd49m:icular day in Athens.
The table below shows some of his findings.

Based on his observations, Alexandros concludes that the temperature, T, in $^{\circ}C$, during the day can be modelled by the equation
$T​=asin(b(t−c))+d$,0≤t≤24,​
where t is the elapsed time, in hours, since midnight.
1.Write down the value of:
(1)a;   
(2)d.   
2.Find the value of b. b ≈   
3.Find the smallest possible value of c, given c>0. c =   
4.Find the time, in hours and minutes, when the temperature first reaches $35^{\circ}C$ during the day.

  Farmer Jimmy bought 50 goats and brought them to his farm. He determined j1b k-zq98xsc e0i(ka9, ormf vhk i:,that 10 of these new goats have a viral diseas9bikxka h s(91 m:kj ,oc0r-,zvi qef8e. The total number of goats on Jimmy's farm is now 600.
The spread of the viral disease on Jimmy's farm is modelled by the logistic function
S(d)​=$\frac{M}{1+Ce^{-kd}}$​,​where S(d) is the number of infected goats after d epidemic days.
1.(1)Write down the value of M. M =   
(2)Find the value of C. C =   
2.Given that quarter of the goats are infected after 12 days, find the value of k, giving your answer correct to 4 significant figures. k ≈   
3.Find the number of days it takes for 90% of the goats to get infected. d ≈    days
4.Draw the graph of y=S(d) on the axes below, indicating clearly the point of inflexion and any asymptotes.

  John operates a small clothing factory that manufactures jeans. John obp7 *2y o j 44zga44jwwz zx*fvyd j4app1b;9hlserves that the weekly total production cost, C, in Australian dollars (AUD), and the number of jeans produced per week, N, can be j pbpp4awjy4lz4g;24zj*yo9w71z v4* hxaf drelated by the equation
C=a$N^b$+K,
where a,b and K are positive constants.
John estimates that the weekly total fixed cost of operatingoperating the factoryfactory is 7500 AUD.
1.Write down the value of K.
After analysing the financial accounting records of a particular month, John finds the data given below.   

2.Draw a scatter diagram of ln(C−K) versus lnN, scaling and shifting the axes if needed.
3.State the type of model that best fits the data displayed on your scatter diagram from part (b).
4.Write down the equation of the regression line of ln(C−K) on lnN.
5.Hence find the value of a and the value of b.
John wants to increase the production rate of jeans up to 1000 pairs per week. a =    b =   
6.Using John's equation, estimate the weekly total cost of producing 1000 jeans.
7.State whether it is valid to use John's equation to estimate the weekly total cost of producing 1000 jeans. Give a reason for your answer. C =    AUD
8.(1)Describe how the data must be entered into your G.D.C. to determine John's equation using power regression method.
(2)Hence verify your answers to part (e).

  Sandra operates a pizzeria that has 24 hours qj 7+cx4x,cw9g;h1j;iey dj adelivery service. Sandra thinks that the daily total production cost, C, in euros (EUR), and the number of pizzas ,q;ej4w 1g+dc hx 7xj ;ac9yjimade per day, N, can be related by the equation
C=aNb+K,where a,b and K are positive constants.
Sandra estimates that the daily total fixed cost of operatingoperating the pizzeriapizzeria is 1000 EUR.
1.Write down the value of K.
After analysing the bookkeeping records of a particular working week, Sandra finds the data given below.   

2.Draw a scatter diagram of $log_2​(C−K)$ versus $log_2​N$, scaling and shifting the axes if needed.
3.State the type of model that best fits the data displayed on your scatter diagram from part (b).
4.Write down the equation of the regression line of $log_2​(C−K)$ on $log_2​N$.
5.Hence find the value of a and the value of b.
Sandra wants to increase the selling rate of pizzas up to 800 items per day. a =    b =   
6.Using Sandra's equation, estimate the daily total cost of producing 800 pizzaspizzas. C ≈    EUR
7.State whether it is valid to use Sandra's equation to estimate the daily total cost of producing 800 pizzas. Give a reason for your answer.
8.(1)Describe how the data must be entered into your G.D.C. to determine Sandra's equation using power regression method.
(2)Hence verify your answers to part (e).