Modelling with Functions (DP IB Maths: AI HL)

Exam Questions

4 hours30 questions
1a
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1 mark

The total cost, C, in New Zealand dollars (NZD), of a premium gym membership at Cityfitness can be modelled by the function

C equals 16.99 t plus 49 comma space space space space space space space space space space space space space space t greater or equal than 0

where t is the time in weeks.

Calculate the cost of the gym membership for a year. Give your answer correct to decimal places.

1b
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2 marks

Find the number of weeks it takes for the total cost to exceed 2000 space NZD.

1c
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1 mark

At Les Mills the initial payment is 20 NZD lower than Cityfitness, however the weekly cost is 8.51 NZD higher than Cityfitness

Write a cost function for a gym membership at Les Mills using an appropriate model.

1d
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3 marks

Calculate how many weeks it will take for the cost of a Les Mills gym membership be more than the cost of a Cityfitness gym membership.

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2a
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4 marks

The front view of the edge of a water tank is drawn on a set of axes below. The edge is modelled by y equals a x squared plus c.

q2a-2-3-medium-ib-ai-sl-maths

Point P has coordinates left parenthesis negative 4 comma 4 right parenthesis comma spacepoint O has coordinates left parenthesis 0 comma 0 right parenthesis spaceand point Q has coordinates left parenthesis 4 comma 4 right parenthesis.

(i)
Find the value of c.

(ii)
Find the value of a.

(iii)
Hence, write down the equation of the quadratic function which models the edge of the water tank.
2b
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2 marks

Given that 1 unit represents 1 m, find the width of the water tank  when its height is 2.25 space straight m.

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3a
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2 marks

The number of German words, W, that Helen remembers after completing a German language course decreases exponentially over time when she does not practice her German. This decrease can be modelled by the function

W left parenthesis t right parenthesis equals a cross times b to the power of negative t end exponent plus 320 comma space space space space space space space space space space space space space space space space space space t greater or equal than 0

Where a and b are positive constants and is the time in years since Helen completed the German language course.

Helen can remember 2400 German words as soon as she completes the German language course.

Find the value of a.

3b
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3 marks

After 2 years Helen has not practiced her German and can only remember 1020 German words.

Find the value of b.

3c
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1 mark

The number of German words Helen remembers never decreases below a certain number of words, c.

Find the value of c.

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4a
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1 mark

In a trial for a new drug, scientists found that the amount of the drug in the bloodstream decreased over time, according to the model

D left parenthesis t right parenthesis equals 1.4 cross times 0.77 to the power of t comma space space space space space space space space space space space t greater or equal than 0

where D is the amount of the drug in the bloodstream in mg per litre left parenthesis m g space L to the power of negative 1 end exponent right parenthesis and t is the time in hours.

Write down the amount of the drug in the bloodstream at t equals 0.

4b
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2 marks

Calculate the amount of the drug in the bloodstream after four hours.

4c
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3 marks

Calculate the time, in hours, for the amount of the drug in the bloodstream to decrease to 0.22 space mg space straight L to the power of negative 1 end exponent.

4d
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2 marks

The scientists found that some of the test subjects had an elevated heart rate for 45 minutes after ingesting the drug.

Find the amount of the drug in the bloodstream when the heart rates from the effected test subjects returned to normal.

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5a
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1 mark

The number of bacteria in a Petri dish is modelled by the function

N left parenthesis t right parenthesis equals 75 cross times 2 to the power of 0.5 t end exponent comma space space space space space space space space space space space t greater or equal than 0

where N is the number of bacteria and t is the time in hours.

Write down the number of bacteria in the Petri dish at t equals 0.

5b
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2 marks

Calculate the number of bacteria present after 10 space hours.

5c
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3 marks

Calculate the time, in hours, for the number of bacteria to reach 10 space 000.

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6a
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1 mark

A remote-controlled sailboat’s velocity is dependent on the wind speed. The sailboat’s velocity is lower during very high and very low wind speeds.

The sailboat’s velocity can be modelled by the function

V left parenthesis w right parenthesis equals 0.0025 w left parenthesis 2 minus w right parenthesis left parenthesis w minus 35 right parenthesis comma space space space space space space space space space 2 less or equal than w less or equal than 35

where V is the sailboat’s velocity, in km space straight h to the power of negative 1 end exponent, and w is the wind speed, in km space straight h to the power of negative 1 end exponent.

Find the sailboat’s velocity when the windspeed is 20 space km space straight h to the power of negative 1 end exponent.

6b
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2 marks

Find the windspeed when the sailboat’s velocity is 5.94 space k m space h to the power of negative 1 end exponent.

6c
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2 marks

Show that V left parenthesis w right parenthesis equals negative 0.0025 w cubed plus 0.0925 w squared minus 0.175 w.

6d
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3 marks

Using your graphics display calculator find the maximum velocity of the sailboat and the windspeed required for this.

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7a
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4 marks

A Ferris wheel rotates at a constant speed, the height of a particular seat above the ground is modelled by the function

H left parenthesis t right parenthesis equals negative 14 square cos space cos space left parenthesis 10 degree cross times t right parenthesis space space plus 16 comma space space space space space space space space space space space t greater or equal than 0 

where H is the height of the seat above the ground, in metres, and t is the elapsed time, in seconds, since the start of the ride.

Write down

(i)
the minimum height of the seat

(ii)
the maximum height of the seat.
7b
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2 marks

Calculate the number of seconds it takes for the Ferris wheel to do one full rotation.

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8a
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1 mark

The water depth, D, in metres, at a port can be modelled by the function

D left parenthesis t right parenthesis equals 5 space sin space sin space left parenthesis 30 degree cross times t right parenthesis space space plus 15 comma space space space space space space space space space space space space space space 0 less or equal than t less or equal than 24

where t is the elapsed time, in hours, since midnight.

8b
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2 marks

The cycle of water depths repeats every P hours. Find the value of P.

8c
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4 marks
(i)
Calculate the maximum and minimum depths.

(ii)
Find the times at which the maximum and minimum depths occur during the day.

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9a
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3 marks

A rectangular sheet of cardboard 60 cm by 100 cm has square sides of x cm cut from each corner. It is folded to make an open box as shown.

q9a-medium-2-3-ib-ai-sl-maths

Show that the volume of the box can be modelled by the function

V equals 4 x cubed minus 320 x squared plus 6000 x.

9b
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2 marks

State the domain of V.

9c
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2 marks

Using your graphics display calculator find the maximum value of V and the value of x which gives this volume 

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10a
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3 marks

Grace leaves a cup of hot tea to cool and measures its temperature every minute. Her results are shown in the table below.

Time, t (minutes) 0 1 2 3 4
Temperature, y (°C) 88 58 43 35.5 k

Write down the decrease in temperature of the tea

(i)
during the first minute

(ii)
during the second minute

(iii)
during the third minute.
10b
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2 marks

Assuming the pattern in the answers to part (a) continues, find the value of k. Give your answer correct to 2 decimal places.

10c
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2 marks

The function that models the change in temperature of the tea is y equals a left parenthesis 2 to the power of negative t end exponent right parenthesis plus b, where b represents the temperature the tea tends towards and a plus b spaceis the initial temperature.

Write down two equations relating a and b.

10d
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2 marks

Find the value of a and b.

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1a
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3 marks

A fence of length L is made to go around the perimeter of a rectangular paddock that borders a straight river. The cost of the fence along the river is $ 15 per metre, while on the other three sides the cost is $ 10 per metre. The total cost of the fence is $ 2000.

Calculate the maximum area of the paddock.

1b
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3 marks

Using the value for the area from part (a), calculate

(i)
the side lengths

(ii)
the total length L spaceof the fence.

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2a
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2 marks

Henry is about to start a 25-game rugby season for his school. Teams receive 4 points for a win and 2 points for a draw. No points are awarded for a loss.

(i)
Using W, D and L as the number of wins, draws and losses respectively, write down an inequality relating the outcomes to the number of games played.

(ii)
Explain why W greater or equal than 0 comma space D greater or equal than 0  and L greater or equal than 0  must also be conditions related to the problem.
2b
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1 mark

Write down an equation for the number of points, P, a team receives from straight W wins and straight D draws.

2c
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2 marks

After 10 games Henry’s team has lost 1 game and they have 32 points.

Find the number of wins and draws Henry’s team has had.

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3a
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1 mark

Antibiotics A and B are applied to a pure culture of bacteria. The number of bacteria present initially for both antibiotics is 6000. The number of bacteria present for antibiotic A, straight N subscript straight A, can be modelled by the function

straight N subscript straight A left parenthesis t right parenthesis equals a cross times b to the power of negative t end exponent comma space space space space space space space space space t greater or equal than 0,

where t is the elapsed time, in hours, since the start of the experiment.

Find the value of a.

3b
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2 marks

The number of bacteria present for antibiotic A after two hours is 2160.

Find the value of b. Give your answer as a fraction.

3c
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3 marks

The number of bacteria present for antibiotic B after four hours is 1185. The number of bacteria present for antibiotic B can be modelled using a similar function to antibiotic A.

Write down the function N subscript B left parenthesis t right parenthesis .

3d
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2 marks

Determine which antibiotic is more effective. Give a reason for your answer.

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4a
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2 marks

In 1967 a house was bought for $10 000.  In 2021 the same house was sold for $800 000.

The average annual growth percentage of the house from 1967 to 2021 is used to form the following model to estimate the value of the house.

straight V left parenthesis t right parenthesis equals a cross times b to the power of t comma space space space space space space space space space space space t greater or equal than 0 comma

where t is the time in years.

Find the value of a and b and write down the function straight V left parenthesis t right parenthesis .

4b
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4 marks

In 2026 the house is bought for $1 300 000.

Calculate the percentage error between the actual value of the house and the estimated value approximated by the model.

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5a
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2 marks

A factory produces cardboard boxes in the shape of a cuboid, with a fixed height of 25 cm and a base of varying area. The area, straight A, of each base can be modelled by the function

straight A left parenthesis x right parenthesis equals x left parenthesis 50 minus x right parenthesis comma space space space space space space space space 10 less or equal than x less or equal than 40 comma

where  is the width of the base of the cardboard box in centimetres.

Cardboard box M has a width of 12 cm.

Find the volume of cardboard box M.

5b
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3 marks

Find the possible dimensions of a cardboard box with volume of 15 space 400 space cm cubed.

5c
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3 marks
(i)
Find the value of x that makes the volume of the cardboard box a maximum.

(ii)
Write down the maximum volume of the cardboard box.

(iii)
State the mathematical shape of the carboard box when its volume is a maximum.

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6a
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1 mark

The downward speed, V, in metres per second, of a bird making a dive into the water to catch a fish can be modelled by the function

straight V left parenthesis t right parenthesis equals 62 minus 62 cross times 3.5 to the power of negative t over 2 end exponent comma space space space space space space t greater or equal than 0,

where t, in seconds, is the time the bird is diving.

Write down the downward speed of the bird at t equals 0.

6b
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2 marks

Determine the equation of the horizontal asymptote for the graph of straight V left parenthesis t right parenthesis.

6c
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2 marks

The bird’s downward speed when it reaches the surface of the ocean is 14 space ms to the power of negative 1 end exponent.

Find the birds downward speed in kilometres per hour left parenthesis km space straight h to the power of negative 1 end exponent right parenthesis.

6d
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3 marks

Find the time, in seconds, for which the bird was diving.

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7a
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1 mark

The temperature, T, of a cake, in degrees Celsius, degree straight C, can be modelled by the function

T left parenthesis t right parenthesis equals a cross times 1.17 to the power of negative t over 4 end exponent plus 18 comma space space space space space space space space space space space t greater or equal than 0 comma

where a is a constant and t is the time, in minutes, since the cake was taken out of the oven.

In the context of this model, state what the value of 18 represents.

7b
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2 marks

The cake was 180 degree straight C when it was taken out of the oven.

Find the value of a.

7c
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2 marks

Find the temperature of the cake half an hour after being taken out of the oven.

7d
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3 marks

The cake is best eaten when its temperature is 75degree straight C to 95degree straight C.

Calculate for how many minutes the cake’s temperature is within this range.

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8a
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2 marks

A hot winter soup has just been removed from the stove and is left outside to cool.  The soup’s temperature can be modelled by the function

straight T left parenthesis t right parenthesis equals a plus b left parenthesis k to the power of negative t end exponent right parenthesis comma space space space space space space space space space space space space space t greater or equal than 0 comma

where t is the time, in minutes, since the soup was removed from the stove.

The temperature outside is 12degree straight C.

Write down the value of a and explain why it has this value.

8b
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2 marks

Initially the temperature of the soup is 95 degree straight C.

Find the value of b.

8c
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2 marks

After two minutes the temperature of the soup is 60degree straight C.

Find the value of k.

8d
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2 marks

After 15 minutes the soup is put into the fridge.

Calculate the temperature of the soup when it is put into the fridge.

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9a
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2 marks

The approximate quantity, , of a decaying radioactive substance can be modelled by the function

N left parenthesis t right parenthesis equals N subscript 0 cross times 2 to the power of negative t over t subscript h end exponent comma space space space space space space space space space space space space t greater or equal than 0 comma

where N subscript 0 is the initial quantity and t is the elapsed time, in years. t subscript h is the half-life of the substance and is a measure of how many years it takes for the quantity of the radioactive substance to decrease by half.

Radioactive substance A has t subscript h equals 1.5 and after 2 years the quantity remaining is 124.

Find N subscript 0. Give your answer to the nearest integer.

9b
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2 marks

Calculate the approximate quantity of the radioactive substance after five years. Give your answer to the nearest integer.

9c
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2 marks

After five years the quantity remaining of the radioactive substance is 25.

Calculate the percentage error between your approximate value found in part (b) and the exact value given above.

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10a
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1 mark

Deserts are known for having high daily temperature ranges. Erica monitors the temperature, in degree straight C, on a particular day in a desert. The table below shows some of the information she recorded.

 

Temperature

Time

Maximum

 41.3 degree straight C

 

Minimum

 0.9 degree straight C  2 colon 00 space am

Erica uses her observations to form the following model for the temperature, straight T degree straight C, during the day

straight T left parenthesis t right parenthesis equals a space cos open parentheses b open parentheses t plus 10 close parentheses close parentheses plus d comma space space space space space space space space space space space space space space 0 less or equal than t less or equal than 24,

where t is the elapsed time, in hours, since midnight.

Calculate the value of t when the maximum temperature occurs and fill in the time in the table above in am/pm format.

10b
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4 marks

Find the values of

(i)
a.

(ii)
b.

(iii)
d.
10c
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3 marks

Erica goes exploring in the desert at 6:30 am and leaves once the temperature reaches 32 degree straight C.

(i)
Calculate the temperature range Erica experiences whilst in the desert.

(ii)
Find the time Erica leaves the desert. Give your time to the nearest ten minutes.

 

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11a
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1 mark

John is a contracted carpenter who earns an annual salary of $55 000. Additionally, John sells innovative wooden art at a market. He goes to the market twice a month and he generally earns $450 from the market.

Estimate John’s total annual income.

11b
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2 marks

John’s actual total annual income over the year is $68 000.

Calculate the percentage error between your answer in part (a) and John’s actual total annual income.

11c
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3 marks

The following table shows different tax rates for residents of the country John lives in.

Income thresholds, in € space left parenthesis x right parenthesis Rate, % Tax payable on this income
x less or equal than 9500

0

0
9500 less than x less or equal than 60   000

15

15 percent sign of amounts over € 9 space 500
60   000 less than x less or equal than 275   000

25

€ 7575 plus 25 percent sign of amounts over € 60 space 000
275   000 less than x

40

€ 157 space 575 plus 40 percent sign of amounts over 275 space 000

Calculate the amount of tax John has to pay. Give your answer to the nearest euro (€).

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1a
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2 marks

Phillip is kayaking to an island located 7 km from where he starts. The distance, D, in kilometres, that Phillip has travelled from his original position can be modelled by the exponential function

D left parenthesis t right parenthesis equals a plus b open parentheses k to the power of negative t end exponent close parentheses comma space space space space space space space space space space space space space space space space space space t greater or equal than 0

where t is the time in minutes since Phillip started kayaking.

State the value of a and explain what the value represents in the context of this question.

1b
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2 marks

Find the value of b.

1c
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2 marks

After 15 minutes Phillip has travelled 5.5 km from his original position.

Find the value of k .

1d
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2 marks

When Phillip is less than 20 m away from the island he can stand up and walk his kayak ashore.

Calculate the time it takes Phillip before he stands up and walks ashore. Give your answer to the nearest minute.

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2a
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2 marks

The average temperature of a city, C, in degrees Celsius, fluctuates throughout a year and can be modelled by the function

C left parenthesis t right parenthesis equals a sin left parenthesis k t right parenthesis plus b comma

where t is the elapsed time, in weeks, since the start of the year.

The average temperature of the city in week 4 is 27 degrees Celsius and in week 28 it is 12 degrees Celsius.

Find the value of k, assuming there are 52 weeks in a year.

2b
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3 marks

Write down two equations connecting  a and b and find their values.

2c
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3 marks

A restaurant stores food in a storage unit outside and when the average temperature of the city gets below 0 degrees Celsius they have to be careful about some things freezing.

Calculate how many weeks of the year that they have to be careful about the food freezing. Give your answer to the nearest integer.

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3a
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4 marks

Algae in a lake can grow exponentially until the lake is fully covered in algae.

Find the number of days it takes for a lake to be fully covered in algae when 1 over 2048 of the lake is covered today and the covered area doubles once every five days.

3b
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4 marks

Find the number of days it takes for a lake to be fully covered in algae when 99.9 percent sign of the lake is uncovered and the covered area increases by a factor of 10 within each 10-day period.

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4
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6 marks

A fence of length straight L, in metres, is made to form a rectangle around a house that borders a forest on one side. The fence does not run along the side next to the forest.  The cost of the fence is $ 22.20 per metre. The total cost of the fence is $ 2250..

Calculate

(i)
the maximal area of the rectangle.

(ii)
the side lengths for the maximal area.

(iii)
the total length of the fence.

 

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5a
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2 marks

A potato top pie is removed from the oven and is left to cool. The pie’s temperature, T, in degree straight C, can be modelled by the function

T left parenthesis t right parenthesis equals a plus b left parenthesis k to the power of negative t end exponent right parenthesis comma space space space space space space space space space t greater or equal than 0 comma

where t is the time, in minutes, since the pie was removed from the oven.

The temperature of the kitchen is 24 degree straight C.

Write down the value of a and explain what it represents in the context of this question.

5b
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1 mark

Initially the temperature of the pie is 200 space degree straight C

Find the value of b.

5c
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1 mark

After five minutes the temperature of the pie is 107 degree straight C.

Find the value of k.

5d
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4 marks

Bacteria in the pie can grow rapidly when its temperature is in the “danger zone” which is between 5 space degree straight C  and space 60 space degree straight C. Food should never be left in the “danger zone” for more than 2 hours. Hence, the pie is put in the fridge after it has been in the “danger zone” for an hour and 20 minutes.

Calculate the total amount of time between the pie being removed from the oven and being put in the fridge. Give your answer to the nearest minute.

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6a
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1 mark

Matt throws a discus in a competition, and its flight can be modelled by the function

y equals fraction numerator x left parenthesis 22 minus x right parenthesis over denominator 20 end fraction plus 1.8 comma space space space space space space space space space space space space space space space space 0 less or equal than x less or equal than 23.5,

where x is the horizontal distance in metres from where the athlete threw the javelin and y is the height of the javelin above the ground in metres.

In the context of the model, explain the significance of the 1.8.

6b
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3 marks

Sketch a graph of the model, labelling any intersections with the coordinate axes and the maximum point.

6c
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3 marks

Matt has been training for this competition for 6 months and his improvement can be modelled by the function

D left parenthesis t right parenthesis equals a ln left parenthesis t plus 1 right parenthesis plus b comma space space space space space space space space space space space space 0 less or equal than t less or equal than 6 comma

where D is the distance of his personal best throw, in metres, and t is the time that has elapsed since he started training, in months and a and b are constants.

At the start of his training Matt’s personal best throw is 14 m. Matt’s throw in the competition was a new personal best.

Find the values of a space and space b.

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7a
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3 marks

A tunnel is being constructed and its opening can be modelled by the quadratic function

h left parenthesis x right parenthesis equals a x left parenthesis b minus x right parenthesis comma space space space space space space space space space space space space space space space space x greater or equal than 0 comma space

where h is the height of the tunnel, in metres, and x is the width of the tunnel, in metres.

It is given that h left parenthesis 10 right parenthesis equals 10 and h left parenthesis 20 right parenthesis equals 15.

Find the values of a and b.

7b
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4 marks

The height required for a lane of traffic is 5 m and each lane requires a width of 2.8 m.

Find the number of lanes of traffic the tunnel can fit.

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8a
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3 marks

A company sells 55 cars per month for a sale price of $2000, whilst incurring costs for supplies, production and delivery of $890 per car. Reliable market research shows that for each increase (or decrease) of the sale price by $50 the company will sell 5 cars less (or more) and vice versa.

Find an expression for the total profit, P, in terms of the sale price, x.

8b
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2 marks

Find the values of x when straight P left parenthesis x right parenthesis equals 0 and explain their significance in the context of the question.

8c
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3 marks

Calculate

(i)
the maximum monthly profit, giving your answer to the nearest dollar.

(ii)
the sale price needed to generate the maximum monthly profit.

(iii)
the number of cars sold to generate the maximum monthly profit.

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9a
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1 mark

A company sells L litres of water per month and their total monthly profit, P, can be modelled by the function

P left parenthesis x right parenthesis equals left parenthesis x minus 0.45 right parenthesis cross times N left parenthesis x right parenthesis comma

where x is the sale price of each litre sold, in dollars, at and N is the linear function for the number of litres the company can sell per month at each given sale price.

In the context of the question, explain the significance of the 0.45.

9b
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2 marks

It is given that N left parenthesis 0.5 right parenthesis equals 400 and N left parenthesis 1.25 right parenthesis equals 250.

Write down the function of N, in the form N left parenthesis x right parenthesis equals m x plus c, where m and c are constants.

9c
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2 marks

Find the values of x when straight P left parenthesis x right parenthesis equals 0 and explain their significance in the context of the question.

9d
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3 marks

Calculate

(i)
the maximum monthly profit.

(ii)
the sale price needed to generate the maximum monthly profit.

(iii)
the number of litres sold to generate the maximum monthly profit.

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