Modelling with Functions (DP IB Applications & Interpretation (AI): HL): Exam Questions

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

$C=16.99t+49,t\ge 0$

where $t$ is the time in weeks.

Calculate the cost of the gym membership for a year. Give your answer correct to 2 decimal places.

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

At Gym B, the initial payment is 20 NZD lower than at Gym A, but the weekly cost is 8.51 NZD higher than at Gym A.

Write down a cost function for a gym membership at Gym B, using an appropriate model.

Calculate how many weeks it will take for the cost of a Gym B membership to be more than the cost of a Gym A membership.

The front view of the edge of a water tank is drawn on a set of axes below. The edge is modelled by $y=a{x}^2+c$.

Point P has coordinates $\left(-4,4\right)$, point O has coordinates $\left(0,0\right)$ and point Q has coordinates $\left(4,4\right)$.

(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.

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

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

$W\left(t\right)=a\times b^{-t}+320,t\ge 0$

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

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

Find the value of $b$.

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

Find the value of $c$.

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

$D\left(t\right)=1.4\times 0.{77}^t,t\ge 0$

where $D$ is the amount of the drug in the bloodstream in mg per litre (${\text{mg L}}^{-1}$) and $t$ is the time in hours.

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

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

Calculate the time, in hours, for the amount of the drug in the bloodstream to decrease to 0.22 ${\text{mg L}}^{-1}$.

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

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

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

$N\left(t\right)=75\times 2^{0.5t},t\ge 0$

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 $t=0$.

Calculate the number of bacteria present after 10 hours.

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

The velocity of a remote-controlled sailboat depends on the wind speed. The velocity of the sailboat is lower when the wind speed is very high or very low.

The velocity of the sailboat can be modelled by the function

$V\left(w\right)=0.0025w\left(2-w\right)\left(w-35\right),2\le w\le 35$

where $V$ is the velocity of the sailboat, in ${\text{km h}}^{-1}$, and $w$ is the wind speed, in ${\text{km h}}^{-1}$.

Find the velocity of the sailboat when the wind speed is 20 ${\text{km h}}^{-1}$.

Find the possible wind speeds when the velocity of the sailboat is 5.94 ${\text{km h}}^{-1}$.

Show that $V\left(w\right)=-0.0025w^3+0.0925w^2-0.175w$.

Use your graphic display calculator to find the maximum velocity of the sailboat and the wind speed at which it occurs.

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

$D\left(t\right)=5\sin \left(30°\times t\right)+15,0\le t\le 24$

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

Write down the depth of the water at midnight.

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

(i) Calculate the maximum and minimum depths.

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

A rectangular sheet of cardboard measures 60 cm by 100 cm. A square of side $x$ cm is cut from each corner, and the sheet is folded to make an open box, as shown.

Show that the volume of the box, in ${\text{cm}}^3$, can be modelled by the function

$V=4x^3-320x^2+6000x$.

State the domain of $V$.

Use your graphic display calculator to find the maximum value of $V$ and the value of $x$ which gives this volume.

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

Write down the decrease in temperature of the tea

(i) during the first minute

(ii) during the second minute

(iii) during the third minute.

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

The temperature of the tea can be modelled by the function $y=a\left(2^{-t}\right)+b$, where $b$ is the temperature that the tea tends towards and $a+b$ is the initial temperature.

Write down two equations relating $a$ and $b$.

Find the value of $a$ and $b$.

Finn borrows $3200 from his parents and decides to pay them back $c$ dollars in the first month and then $m$ dollars each subsequent month.

After two months Finn has paid back his parents a total of $1000, this can be expressed as $m+c=1000$. After half a year he still owes his parents $1000.

Find another equation connecting $m$ and $c$.

Find the value of m and c.

Finn's parents add a charge of 6.25% of the $3200 to the amount he owes.

Calculate the number of months it takes Finn to pay back his parents.

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, $A$, of each base can be modelled by the function

$A\left(x\right)=x\left(50-x\right),10\le x\le 40$,

where $x$ 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.

Find the possible dimensions of a cardboard box with a volume of $15 400{\text{ cm}}^3$.

(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.

A company sells water, and its total monthly profit, $P$, can be modelled by the function

$P\left(x\right)=\left(x-0.45\right)\times N\left(x\right)$,

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

Interpret, in context, the value 0.45.

It is given that $N(0.5)=400$ and $N(1.25)=250$.

Write down the function of $N$, in the form $N\left(x\right)=mx+c$, where $m$ and $c$ are constants.

Find the values of $x$ when $P\left(x\right)=0$ and interpret, in context, these values.

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.

A skydiver jumps from an aircraft at a point directly above a fixed point, $\mathrm{O}$, on the ground. The skydiver's path can be modelled by

$h\left(x\right)=3200-0.5x^2$

where $h$ is the height of the skydiver above the ground, in metres, and $x$ is the horizontal distance of the skydiver from $\mathrm{O}$, in metres.

(i) Explain the meaning of the value $3200$ in this model.

(ii) Find the horizontal distance from $\mathrm{O}$ at which the skydiver lands.

Sketch the graph of $h$ against $x$ for $0\le x\le 80$.

Explain why the model is not suitable for values of $x$ greater than 80 m.

A golf ball is hit from a point $\mathrm{O}$ on a horizontal golf course. Its path can be modelled by

$y=-\frac{x^2}{180}+x$

where $x$ metres and $y$ metres are the horizontal and vertical displacements of the ball from $\mathrm{O}$. Upwards is taken to be the positive vertical direction.

Write down the set of values of $x$ for which the model is valid. Give a reason for your answer.

Sketch the graph of $y$ against $x$ for the values of $x$ found in part (a), labelling the points where the graph meets the $x$-axis.

Find the maximum height reached by the ball.

The average fat-free mass, $M$ kg, of footballers aged $a$ years can be modelled by the function

$M\left(a\right)=10{\log }_{10}\left(a-15\right)+50$, for $16\le a\le 25$.

Calculate the average fat-free mass of footballers aged

(i) 16 years

(ii) 25 years.

A linear model, $L$, passes through the two points found in part (a).

Find an expression for $L\left(a\right)$.

Use $L$ to estimate the average fat-free mass of a footballer aged 20 years. Calculate the percentage error in this estimate, taking the value given by $M$ as the exact value.

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

$N\left(t\right)=N_0\times 2^{-\frac{t}{t_h}},t\ge 0$

where $N_0$ is the initial quantity and $t$ is the elapsed time, in years. $t_h$ 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 $t_h=1.5$ and after 2 years the quantity remaining is 124.

Find $N_0$. Give your answer to the nearest integer.

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

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.

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

$V\left(t\right)=62-62\times 3.5^{-\frac{t}{2}},t\ge 0$

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

Write down the downward speed of the bird at $t=0$.

Write down the equation of the horizontal asymptote of the graph of $V\left(t\right)$.

The bird’s downward speed when it reaches the surface of the ocean is 14 ${\text{m s}}^{-1}$.

Find the bird’s downward speed in kilometres per hour (${\text{km h}}^{-1}$).

Find the time, in seconds, for which the bird was diving to the ocean's surface.

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 $W\ge 0$, $D\ge 0$ and $L\ge 0$ must also be conditions related to the problem.

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

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.

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 each time he goes.

Estimate John’s total annual income.

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.

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

Calculate the amount of tax John has to pay on his actual total annual income. Give your answer to the nearest euro (€).

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, ${\mathrm{N}}_{\mathrm{A}}$, can be modelled by the function

${\mathrm{N}}_{\mathrm{A}}\left(t\right)=a\times b^{-t},t\ge 0$,

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

Find the value of $a$.

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

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

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.

Find an expression for $N_B\left(t\right)$.

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

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.

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

(i) the side lengths

(ii) the total length $L$ of the fence.

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.

$V\left(t\right)=a\times b^t,t\ge 0$

where $t$ is the number of years since 1967.

Find the values of $a$ and $b$, and write down the function $V\left(t\right)$.

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.

The temperature, $T$, of a cake, in degrees Celsius (°C), can be modelled by the function

$T\left(t\right)=a\times 1.{17}^{-\frac{t}{4}}+18,t\ge 0$

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.

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

Find the value of $a$.

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

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

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

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

$T\left(t\right)=a+b\left(k^{-t}\right),t\ge 0$

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

The temperature outside is 12 °C.

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

Initially the temperature of the soup is 95 °C.

Find the value of $b$.

After two minutes the temperature of the soup is 60 °C.

Find the value of $k$.

After 15 minutes the soup is put into the fridge.

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

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

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

$T\left(t\right)=a\cos \left(b\left(t+10\right)\right)+d,0\le t\le 24$

where $t$ is the elapsed time, in hours, since midnight, and $b\left(t+10\right)$ is measured in degrees.

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

Find the values of

(i) $a$.

(ii) $b$.

(iii) $d$.

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

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

$h\left(x\right)=ax(b-x),x\ge 0,$

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

It is given that $h\left(10\right)=10$ and $h\left(20\right)=15$.

Find the values of $a$ and $b$.

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.

A fence of length $L$, 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 total length of the fence.

(ii) the side lengths for the maximal area.

(iii) the maximal area.

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

$D\left(t\right)=a+b\left(k^{-t}\right),t\ge 0$

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.

Find the value of $b$.

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

Find the value of $k$ .

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.

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 $\frac{1}{2048}$ of the lake is covered today and the covered area doubles once every five days.

Find the number of days it takes for a lake to be fully covered in algae when 0.1 % of the lake is covered today and the covered area increases by a factor of 10 every 10 days.

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 fewer (or 5 more) cars.

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

Find the values of $x$ when $P\left(x\right)=0$ and interpret, in context, these values.

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.

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

$C\left(t\right)=a\sin \left(kt\right)+b$

where $t$ is the elapsed time, in weeks, since the start of the year, and $kt$ is measured in degrees.

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.

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

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.

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

$T\left(t\right)=a+b\left(k^{-t}\right),t\ge 0$

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

The temperature of the kitchen is 24 °C.

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

Initially the temperature of the pie is 200 °C.

Find the value of $b$.

After five minutes the temperature of the pie is 107 °C.

Find the value of $k$.

Bacteria in the pie can grow rapidly when its temperature is in the “danger zone” which is between 5 °C and 60 °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.

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

$y=\frac{x\left(22-x\right)}{20}+1.8,x\ge 0$

where $x$ is the horizontal distance in metres from where the athlete threw the discus and $y$ is the height of the discus above the ground in metres. The model applies until the discus hits the ground.

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

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

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

$D\left(t\right)=a\ln \left(t+1\right)+b,0\le t\le 6$

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 $a$ and $b$.