Further Modelling with Functions (DP IB Maths: AI HL)

Exam Questions

4 hours35 questions

4 marks

The fare of a taxi ride starts at $2.50 and increases by $4.15 per km, for the first 5.5 km. After 5.5 km the rate charged per km decreases by $0.50.

Write down a piecewise function that models the fare of the taxi ride and show that the function is continuous when the taxi ride reaches 5.5 km.

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2 marks

Calculate the distance of a taxi ride that costs $15.

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3 marks

On a particular ride, a taxi driver accidentally takes a wrong turn 2.8 km into the ride that extends the journey by 2.1 km. Since the driver cannot switch off the fare calculator in the taxi without losing the details of the ride he decides to calculate the fair price for his customer, by removing the cost from the extra distance travelled. The total length of the journey, including the 2.1 km detour, was 9.4 km.

Calculate the proper fare the customer should pay, having appropriately discounted the distance of the detour.

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2 marks

The average temperature, $T ° C$ , of a city in winter every day can be modelled by the function

$T (t) = a \sin (b t + k π) + d, 0 \leq t < 24,$

where $t$ is the time, in hours after midnight, $a$ and $d$ are constants, and $b$ is measured in radians.

Find the value of $b$ . Give your answer in terms of π.

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3 marks

The average daily temperature range in the city is $6 ° C$ . The average maximum daily temperature is $2 ° C$ and the average minimum daily temperature is at 3:00 am.

Find the values of $a, k and d$ .

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4 marks

Sketch the graph of $T$ and label any intersections with the axes, local maximums and local minimums.

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2 marks

The daily distance covered by an animal depends on the weather, and so varies each month. The following model represents the average daily distance, D km, covered by the animal each month.

$D (t) = a \cos (b t + k) + d, 0 \leq t < 12,$

Where $t$ is the time, in months after the beginning of the year, $a$ and $d$ are constants, and $b$ is measured in degrees.

Find the value of $b$ .

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3 marks

The range of $D$ is $22 km$ , the maximum daily distance covered is $29 km$ and the minimum daily distance covered occurs when $t = 6.5$ .

Find the values of $a, k and d$ .

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4 marks

Sketch the graph of $D$ and label any intersections with the axes, local maximums and local minimums.

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1 mark

Given that the animal covers the most distance in summer, state which hemisphere the animal is likely to live in.

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5 marks

The following piecewise function models the depth of a pool, in metres.

$d (x) = \{\begin{matrix} \frac{1}{2} (x + a)^{2} + b, \\ - 2, \\ \frac{1}{4} x + c, \end{matrix} \begin{matrix} 0 \leq x \leq 2 \\ 2 < x \leq 10 \\ 10 < x \leq d \end{matrix}$

where $x$ represents the horizontal distance in metres from the deep end and $a, b and c$ are constants. The depth at both ends of the pool is 0 m.

Find the values of $a, b$ and $c$ such that $f$ is a continuous function.

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1 mark

Find the value of $d$ .

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1 mark

Write down the maximum depth of the pool and the length of the pool.

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4 marks

A basketball streaming site offers three membership options depending on the number of matches the member wants to watch per week. However, if the member does want to watch more matches in a given week they will be charged per match. The membership options are summarized in the table below.

Membership	Matches/Week	Extra charge/Match	Weekly Cost
Casual	7	$0.75	$6.50
Baller	14	$0.50	$8.50
Premium	Unlimited	-	$11.50

The total weekly cost, $C, for the casual and baller memberships can be modelled as a piecewise linear function, where $m$ is the number of matches watched in a given week. Determine the models for each type of membership.

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3 marks

Find the total weekly cost for each membership if in a given week a member wants to watch 20 games.

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3 marks

The depth, $D$ m, of an underwater sound wave can be modelled by the function

$D (t) = 18 - 3.4 \sin (0.523 t)$

where $t$ is the elapsed time, in seconds, since the first sound wave was detected by the sensor.

Find the minimum and maximum depths of the sound waves as they pass the sensor.

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3 marks

Find the first time after 12 seconds at which the depth of the wave reaches 18.2 m.

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2 marks

The income tax rates for a country are shown in the table below.

Income, $$ x$	Income tax rate, $y$ %
$0 < x \leq 22 000$	0
$22 000 < x \leq 64 000$	18
$64 000 < x \leq 130 000$	22
$x > 130 000$	29

Calculate the amount of tax payable on the first $65 000 of income.

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3 marks

Calculate the income of someone who has $11 520 of income tax payable.

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3 marks

John is paid an annual salary, before tax, of $75 000. He works 10 months of the year and then he decides to take the rest of the year off.

Calculate the amount of tax payable for John.

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2 marks

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

$H (t) = a \sin (b t - c) + d, 0 \leq t \leq 48$

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.

The ride starts from the lowest point on the Ferris wheel and takes a total of 48 seconds.

Find the value of $b$ .

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3 marks

The seat reaches a minimum height of m and a maximum height of 42 m.

Find the values of $a, c$ and $d$ .

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3 marks

Passengers on the Ferris wheel have the best view when their seat is above 25 m.

Calculate the number of seconds for which the passengers have the best view.

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2 marks

A company producing small boats sells 60 boats per month for a sale price of $$ 1200$ , with each boat costing $$ 700$ to produce. Reliable market research suggests that for each increase (or decrease) of the sale price by $$ 50$ the company will sell 10 units less (or more).

Given that $N$ is the number of boats the company can sell per month with a sale price of $$ x$ , show that $N (x) = - \frac{1}{5} x + 300$ .

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3 marks

Given that $P$ is the total monthly profit the company makes from selling the boats for a sale price of $$ x$ , show that $P (x) = - \frac{1}{5} x^{2} + 440 x - 210000$ .

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3 marks

Find the number of boats the company must produce to maximise monthly profit, given that the maximum monthly production is

(i)

75 boats

(ii)

115 boats.

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1 mark

Write down two intervals of $x$ for which the company makes a loss and state an economic reason why for each interval.

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10a

2 marks

For a particular type of coffee, a typical mug contains 100 mg of caffeine. The half-life of the amount of caffeine in the bloodstream is 2.5 hours.

Assuming the 100 mg of caffeine from a mug of coffee is absorbed immediately after drinking it, the amount of caffeine, $C$ mg, left in the bloodstream $t$ hours after consumption can be modelled by the equation

$C = A e^{- k t}$

where $A$ and $k$ are positive constants.

Write down the value of $A$ .

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10b

2 marks

Find the value of $k$ , giving your answer to 2 significant figures.

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10c

3 marks

Find the amount of caffeine in the bloodstream after 6 hours.

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10d

3 marks

A consumer wishes to cut down their caffeine intake and so makes a drink using half the amount of coffee in a mug.

Find the amount of caffeine in the bloodstream for this consumer after 6 hours and state an assumption you have made in finding your answer.

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11a

2 marks

A rhino is raised in a zoo and his height, $h$ metres, is modelled by the logistic function

$h (t) = \frac{L}{1 + 1.9 e^{- 0.27 t}}, t \geq 0,$

where $t$ is the number of years since his birth. The rhino’s height reaches a limit of 1.82 m as he ages.

State the value of $L$ .

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11b

2 marks

Find the rhino’s height on his 12th birthday.

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11c

3 marks

The rhino’s $n$ th birthday is the first birthday in which he is double the value of $h (0)$ .

Find the value of $n$ .

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12a

2 marks

The surface of a large pond is partially covered by algae. A limnologist (a scientist who studies freshwater systems) monitors the area, $A m^{2}$ , of the pond covered by algae, $d$ days, after first discovering its presence.

The limnologist plots a graph of $\log A$ against $d$ , and after 4 days the graph is a straight line passing through the points (0, 1.7) and (4, 1.9).

mi-q12a-2-6-further-modelling-with-functions-ib-ai-hl-medium-maths_dig

The limnologist believes the area of the pond covered by algae can be modelled by the equation $A = A_{0} b^{d}$ .

Find the value of $A_{0}$ , giving your answer to two significant figures, and explain its meaning in the context of the algae covering the lake.

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12b

2 marks

Find the gradient of the straight line.

ii)

Hence find the value of

b

correct to 2 significant figures

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12c

2 marks

Using the rounded values for $A_{0}$ and $b$ in the model predict the area of the pond covered by algae after 20 days.

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13a

2 marks

In a production process the amount of a pollutant, $P$ ppm (parts per million), in the surrounding air seconds after the process began, is monitored. A chemist produces the graph below of the first 10 seconds of the process.

mi-q12-2-6-further-modelling-with-functions-ib-ai-hl-medium-maths_dig

The graph passes through the points $(0, - 0, 7)$ and $(1, 4, 0)$ . The chemist suggests that a model of the form $P = a t^{b}$ , where $a$ and $b$ are constants, can be used to predict the amount of pollutant in the air.

Find the gradient of the graph.

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13b

1 mark

Find the equation of the straight line.

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13c

3 marks

Find the values of $a$ and $b$ .

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13d

2 marks

The process stops after a maximum running time of 20 seconds.

Find the maximum amount of the pollutant produced during one occurrence of the production process. State your answer to 2 significant figures.

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2 marks

Ben is planning an event, so he decides to investigate potential catering companies. Company A quotes a price of $1900 and says they will provide staff for food and drink service at an extra cost of $200 per hour, for a minimum of two hours. Company B quotes a price of $2100 and says they will provide staff for food and drink service for $160 per hour for a minimum of one hour and a maximum of six hours.

Write down two functions, including their domains, for the total revenue generated by company A and company B from catering the event.

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2 marks

Ben has already calculated other costs involving in running the event and has concluded that the event needs to be at least three hours long in order to make a profit.

On the same diagram, sketch the graphs of both of the functions found in part (a).

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3 marks

Determine which catering company Ben should use to minimise the cost based on the duration of the event.

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4 marks

A Ferris-wheel-type attraction at a wildlife park is partially submerged underwater to enable passengers to observe both land and aquatic creatures at the park. The wheel rotates at a constant speed and the height of a viewing capsule above the water level is modelled by the function

$H (t) = 20 \sin (\frac{π}{18} t) + 2, t \geq 0$

where $H$ is the height, in metres, of the viewing capsule, and t is the elapsed time, in minutes, since the start of the ride.

Find

(i) the height above the water level at which passengers enter a viewing capsule,

(ii) the maximum height above the water level that a viewing capsule reaches,

(iii) the maximum depth below the water level that a viewing capsule reaches,

(iv) the amount of time spent on one full rotation on the Ferris-wheel.

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2 marks

Calculate, to one decimal place, the length of time for which a capsule is under water during one ride.

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1 mark

The sound intensity level L, measured in decibels (dB), is given by $L = 10 \log (\frac{I}{I_{0}})$ , where I is the sound intensity and $I_{0}$ is the reference sound intensity.

Write down the sound intensity level, $L$ dB, in the case where $I = I_{0}$ .

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2 marks

Using a reference sound intensity of 20 dB, find the sound intensity level of a sound intensity of 30 dB.

ii)

Find the sound intensity for a sound that has a sound intensity level of 2 dB using a reference sound intensity of 15 dB.

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5 marks

(i)

Show that for a sound intensity level of 70 dB,

I = 10^{7} I_{0}

(ii)

Hence, or otherwise, write down an equation in the form

I = 10^{x} I_{0}

for a sound intensity level of 50 dB.

(iii)

Hence show that a sound intensity (

I

) of 70 dB is 100 times more intense than a sound intensity of 50 dB.

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3 marks

A manufacturing process takes place inside a sealed chamber and produces a pollutant that decays over time. After the process is completed, at time t = 0 seconds, the amount of pollutant, P ppm (parts per million) in the chamber is modelled by

$P = P_{B} + P_{A} e^{k t}$

Write down an expression for

(i)

the background level of the pollutant in the chamber,

(ii)

the amount of pollutant in the chamber at the moment the process completes,

(iii)

the half-life of the pollutant.

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2 marks

The half-life of the pollutant is $t_{0.5} = e^{4}$ .

Find the value of $k$ .

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3 marks

It is safe for the chamber to be opened once the amount of pollutant falls to an amount

that is less than 10% above the background level.

Given that $P_{A} = 5 P_{B}$ , find the minimum number of complete minutes the chamber should remain sealed for after the manufacturing process completes.

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3 marks

Once it has reached a certain height above ground level, the height, h centimetres, of a particular species of sunflower can be modelled by the logistic function

$h (t) = \frac{300}{1 + 199 e^{k t}}, t \geq 0 .$

where $t$ is the number of days after the model becomes applicable and $k$ is a constant.

(i)

Write down the value of the carrying capacity and explain what this represents in terms of the species of sunflower.

(ii)

Write down the height above ground level the sunflower needs to reach so that the model becomes applicable.

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2 marks

Given that $k = 0.1 .$

Show that, according to the model, after 75 days the sunflower has grown to within 10% of its maximum possible height.

ii)

Find the number of days required for the sunflower to reach 60% of its maximum possible height.

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1 mark

According to horticulturists, this species of sunflower takes 80 – 120 days to reach its full height. Give one limitation of the model.

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2 marks

A model of the form $V = A \times 3^{r}$ , where $A > 0$ , is proposed by an internet search engine to estimate the number of visits per day to websites.

Each website is given a rank, $r$ , where $0 \leq r \leq 10$ , based on how popular the search engine company decides that website is.

Write down, in terms of $A$ , the lowest and highest possible visits a website can achieve in a single day according to the model.

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4 marks

Sketch the graph of $\log V$ against $r$ . Label the points corresponding to the answers found in part (a).

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2 marks

Given that the graph of $\log V$ against $r$ passes through the point $(5, 5.08)$ , find the value of $A$ . Give your answer correct to one significant figure.

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2 marks

Show that a website with a rank of 7 would have approximately 80 times as many visits per day as a website with a rank of 3 would have.

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4 marks

The water depth, d m (metres), at a port can be modelled by the function

$d (t) = A \sin (\frac{π}{12} (2 t - 1)) + B, 0 \leq t \leq 24$

where t is the elapsed time in hours since midnight. A and B are constants.

With regards to the function $d (t)$ , and giving your answers in terms of $A$ and/or $B$ as appropriate, write down

(i) the phase shift,

(ii) the period,

(iii) the amplitude,

(iv) the equation of the principal axis.

How did you do?

2 marks

Explain what the period and the amplitude of $d (t)$ mean in terms of the depth of water in the port.

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2 marks

Find the time of day at which the water depth first reaches its minimum value.

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3 marks

All water vehicles are prohibited from entering the port whilst the depth of water is below 3 m.

It is given that $A = 3$ and $B = 5$ .

Find the times of day between which all water vehicles are prohibited from entering the port.

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3 marks

The sketch below shows the graph of $\log y$ against $\log x$ .
The graph passes through the point (0,4) and has a gradient of 1.5.

mi-q8a-2-6-further-modelling-with-functions-ib-ai-hl-hard-maths_dig

Write down a function for $y$ , in terms of $x$ .

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3 marks

The sketch below shows the graph of $y$ against $\log x$ .

The graph passes through the point $(0, 1.5)$ and has a gradient of 0.5.

mi-q8b-2-6-further-modelling-with-functions-ib-ai-hl-hard-maths_dig

Write down a function for $y$ in the form $y = \log f (x) .$ .

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2 marks

The profile of a non-symmetrical skate ramp is modelled by the piecewise function

$h (w) = \{\begin{cases} \frac{3}{w + 1} - \frac{1}{2} & 0 \leq w \leq 5 \\ 0 & 5 \leq w \leq 7 \\ \frac{5}{128} {(w - 7)}^{2} & 7 \leq w \leq 15 \end{cases}$

where $h$ m is the height of the ramp above ground level at the point $w m$ from the southernmost end of the skate ramp.

Verify that both the southernmost and northernmost ends of the skate ramp are at a height of 2.5 m above ground level.

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2 marks

Find the height of the ramp when it is a distance of $3 m$ from the southernmost end of the skate ramp.

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2 marks

Find the other distance from the southernmost end of the skate ramp when the ramp is at the same height as the answer to part (b). Give your answer to a sensible degree of accuracy.

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1 mark

Briefly explain, why, in this context, negative values of $w$ could be considered.

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10a

2 marks

The magnitude of an earthquake, $R$ , on the Richter scale can be modelled by the function

$R (E) = \frac{2 \ln E}{3 \ln 10} - 3.2$

where E is the amount of energy, in joules (J), released by the earthquake.

Find the magnitude, to one decimal place, of an earthquake which releases $7.2 \times 10^{10}$ J of energy.

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10b

2 marks

In February of 2011 an earthquake with magnitude 6.3 struck the region of Canterbury in the South Island of New Zealand.

Find the amount of energy, in joules, released by this earthquake, giving your answer in standard index form, correct to 2 significant figures.

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11a

2 marks

The average time,T in minutes, it takes for people to choose a movie to watch at the cinema can be modelled by the function

$T (n) = 0.72 \log_{5} (n + 1), n \geq 3$

where $n$ is the number of movies being shown at the cinema.

Calculate the number of minutes it would take someone to choose a movie at a cinema that is playing seven movies.

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11b

2 marks

Find an expression for the inverse function $T^{- 1} (n)$ .

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11c

2 marks

It takes 1.073 minutes for someone to choose a movie at a cinema. Find the number of movies being shown at the cinema.

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2 marks

If a substance decays by $k %$ per year, its half-life, H years, is given by

$H (k) = \frac{\ln 0.5}{\ln (1 - \frac{k}{100})}, 0 < k < 100$

Find the annual percentage decay rate of a substance with a half-life of 32.5 years.

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2 marks

Find the half-life of a substance with an annual percentage decay rate of 12.2%.

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3 marks

Americium-241 has a half-life of 432.2 years and a sample initially contains 40 grams.

Find the number of grams in the sample after 55 years.

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3 marks

Consider the function $v$ , where $v$ is a function of $t$ and $1 \leq t \leq 10 .$

Sketches of the graphs of $v$ against $t$ and $\log v$ against $\log t$ , are shown below.

mi-q2a-2-6-further-modelling-with-functions-ib-ai-hl-very-hard-maths-dig

Use the information given in the graph sketches to find $v$ in terms of $t$ .

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3 marks

The number of calculations, $C$ , a supercomputer can perform per second is related to the speed of its processor, s GHz.

The graph of $C$ against $s$ is an exponential curve passing through the point $(4.2 \times 10^{8})$ .

The semi-log graph of $\log C$ against $s$ is a straight line passing through the point $(3, 8)$ .

Find $C$ , as a function of $s$ .

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4 marks

Kerry is organising her local town’s summer festival and wants to hire a bouncy castle for a minimum of 2 days up to a maximum of 12 days, depending on cost. She has two potential suppliers to choose from.

Bounce-o-rama can supply a bouncy castle at $£ 110$ per day for the first three days, $£ 85$ per day for the next three days and £60 for every day thereafter.

Inflate-o-castle can supply a bouncy castle at $£ C$ per day, determined by the equation $C = 163 d - 8 d^{2} - 87$ , where $d$ is the number of days the bouncy castle is hired for, with a limit of 10 days.

Given that Kerry has a budget of between £300 and £750, analyse which supplier she should use across the range of her budget in order to obtain the best value for money

How did you do?

1 mark

State the number of whole days for which both suppliers charge the same amount.

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2 marks

Briefly explain why it would not be in Inflate-o-castle’s interest to offer more than 10 days of bouncy castle hire using the given cost equation.

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2 marks

A company makes a device that can be attached to small items such that if the item were to be dropped into water, the item would float rather than sink. In testing the device, an employee throws the device over the edge of a boat and the height, $h m$ , of the device above the level of calm water $t$ seconds after being thrown is modelled by the function

$h (t) = A e^{- 0.2 t} \sin (2 t + \frac{π}{6}) t \geq 0$

where $A$ is a constant.

Given that the employee throws the keys from a height of above the level of calm water, find the value of A.

How did you do?

3 marks

Find

(i) the maximum height above the calm water level the device reaches,

(ii) the time at which the device first reaches calm water level,

(iii) the maximum depth below the calm water level the device reaches.

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3 marks

Find the times between which the device is deeper than 0.5 m below calm water level.

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1 mark

Describe the motion of the device for large values of $t$ .

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4 marks

A solid substance, with an extremely high melting point temperature, has a volume of $5 {cm}^{3}$ at $0 ° C$ .

The substance expands as it is heated, but has a carrying capacity of $4000 {cm}^{3}$ . The volume of the substance is 0.0001 m³ when heated to $46.5 ° C .$

The volume, $V {cm}^{3}$ at temperature $θ ° C$ , of the substance is modelled by the function

$V = \frac{A}{1 + B e^{- k θ}}$

where $A, B$ and $k$ are positive constants.

Find the values of $A, B$ and $k$ , giving the value of $k$ to two significant figures.

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2 marks

Find the volume, to the nearest whole cubic centimetre, of the substance when it is heated to $80 ° C$ .

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1 mark

Briefly explain the relevance of the substance’s extremely high melting point temperature in the context of the question.

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1 mark

The value, $V$ , measured in thousands of dollars, of a mobile phone initially worth $1000, is modelled by the function

$V = a + b \ln (y + 1) 0 \leq y \leq c$

where the age of the mobile phone is y years and $a, b$ and c are constants.

Write down the value of $a$ .

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3 marks

(i)

Explain what the constant c represents in the context of the model.

(ii)

Given that

b = - \frac{k}{10}

, where

k

is a positive integer, and that

11.1 < c < 11.2

, find the exact value of

b

How did you do?

2 marks

Find the age of the phone when its value has halved from its initial worth.

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1 mark

State an assumption the model makes about the mobile phone.

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5 marks

In a fairground “lift and drop” ride, a row of seated passengers are projected vertically upwards until the ride reaches its maximum height (“the lift”), then the ride returns to its starting position (“the drop”).

The height of a passenger, $h$ cm, above ground level at time t seconds after the ride begins is modelled by the function

$h (t) = a \cos (b (t - c)) + d$

where $a, b, c$ and $d$ are constants.

Given that

passengers are 0.75 m above the ground when they board the ride,

the maximum height a passenger reaches on the ride is 12.25 m,

it takes 24 seconds for the ride to complete one “lift and drop”,

c takes the smallest positive value possible,

find the values of $a, b, c$ and $d$ .

How did you do?

3 marks

(i)

Find the times at which a passenger is 6 m further from the ground than when they first sat on the ride.

(ii)

Find the height of a passenger from the ground after 10 seconds.

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3 marks

A temporary lamppost that lights part of a forest pathway is powered via an outdoor electricity generator. The cost, $£ C$ of running the generator for $h$ hours is modelled by the equation

$C = a \log (b h + 1) h \geq 0$

where $a$ and $b$ are positive constants.

Given that, to three significant figures, the cost of running the generator for 2.31 hours is the half the cost of running the generator for 7.28 hours, find the value of b correct to one decimal place.

How did you do?

2 marks

Given that is an integer and that £1 would run the generator for just under 20 minutes, find the value of $a$ .

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4 marks

Find

(i) the cost of running the generator for 12 hours, to the nearest penny (£0.01),

(ii) the maximum length of time (in hours and minutes) the generator can be (continuously) run for at a maximum cost of £8.

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3 marks

A model of the form $P = P_{0} e^{- k t}$ is proposed to model the amount of a toxic gas, P parts per million (ppm), in a chamber, t seconds after a chemical reaction has taken place inside the chamber. $P_{0}$ and $k$ are positive constants.

(i)

Write down the amount of toxic gas in the chamber at the instant the chemical reaction completes?

(ii)

What can you deduce about the background level of the toxic gas inside the chamber? Explain your answer.

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1 mark

Write down, in terms of $k$ , the exact half-life, $t_{0.5}$ of the toxic gas.

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3 marks

Given that the amount of toxic gas reduces by 15% every 20 seconds, find its half-life to the nearest second.

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2 marks

The chemical reaction also produces a harmless gas which has some background level inside the chamber. The amount of this gas increases with time after the chemical reaction has taken place.

Suggest a model for the amount of harmless gas in the chamber $t$ seconds after the chemical reaction completes, defining any constants used and any restrictions on their values.

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10a

4 marks

The Theoretical Zipf Distribution relates the rank and frequency of words used in the English language.

For a large body of text a word of rank $r$ , appears $f$ times where $f = k r^{- 1}$ , $k$ is the number of times the most used word occurs.

The graph below shows $\log f$ plotted against $\log r$ for a particular novel in which the most used word is “the” and occurs 5623 times.

The graph passes through the points $(1, 2.6)$ and $(3, 0.3)$ .
mi-q10a-2-6-further-modelling-with-functions-ib-ai-hl-very-hard-maths-dig

Use the graph to find $f$ in terms of $r$ and comment on whether the distribution of words in this novel follows the Theoretical Zipf Distribution.

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10b

2 marks

The 8^th ranked word in the novel was “was”. Use your answer to part (a) to estimate how many times “was” was used in the novel.

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11a

3 marks

The profile of a mountain is modelled by the piecewise function

$h (x) = \{\begin{cases} 0.02 x^{2} & 0 \leq x \leq k \\ 0.01 {(x - 16)}^{3} + 3.52 & k < x \leq 20 \end{cases}$

where $h$ km is the height of the mountain at a horizontal distance $x$ km along the ground from its basecamp.

$h (x)$ is a continuous function at the point $x = k$ .

Find the value of $k$ and the height of the mountain at this point.

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11b

2 marks

Find

(i)

the height of the mountain at a horizontal distance of 6 km along the ground from its basecamp,

(ii)

the horizontal distance along the ground from the basecamp at the point where the mountain’s height is 3.82 km.

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11c

3 marks

Find the height and the horizontal distance from mountain’s basecamp at the point where the slope of the mountain is flat.

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