SL 2.6—Modelling skills Questionbank

Practice SL 2.6—Modelling skills with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

SLPaper 2

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 $(-4,4)$ , point $O$ has coordinates $(0,0)$ and point $Q$ has coordinates $(4,4)$ .

Find the value of $c$ and of $a$ .

[2]

Hence, write down the equation of the quadratic function which models the edge of the water tank.

[1]

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

[3]

Question 2

SLPaper 2

The average temperature of a city, $C$ , in degrees Celsius, fluctuates throughout a year and can be modelled by the function $C(t)=a \sin (k t)+b$ 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.

[1]

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

[4]

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

[2]

Question 3

SLPaper 2

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 total profit, $P$ , in terms of the sale price, $x$ .

[2]

Find the values of $x$ when $P(x)=0$ and explain their significance in the context of the question.

[4]

Calculate: 3. maximum monthly profit, giving your answer to the nearest dollar.

[1]

the sale price needed to generate the maximum monthly profit.

[1]

number of cars sold to generate the maximum monthly profit.

[1]

Question 4

SLPaper 2

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.

[6]

Question 5

SLPaper 2

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

	Temperature	Time
Maximum	$41.3^{\circ} \mathrm{C}$
Minimum	$0.9^{\circ} \mathrm{C}$	2:00 am

Erica uses her observations to form the following model for the temperature, $T^{\circ} C$ , during the day $T(t)=a \cos (b(t+10))+d, 0 \leq t \leq 24$ 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.

[2]

Find the value of $a, b, d$ .

[4]

Erica goes exploring in the desert at $6: 30$ am and leaves once the temperature reaches $32^{\circ} \mathrm{C}$ . 3. Calculate the temperature range Erica experiences whilst in the desert.

[2]

Question 6

SLPaper 2

A local bakery offers fresh baguettes for delivery. The total cost to the customer, $C$ in Euros (€), is modeled by $C(n) = 3.2n + 5$ , where $n$ is the number of baguettes ordered and includes a fixed delivery fee.

State what the value of $3.2$ represents.

[1]

State what the value of $5$ represents.

[1]

Write down the minimum number of baguettes that can be ordered.

[1]

Sophie has 30 Euros. Find the maximum number of baguettes Sophie can order.

[3]

Question 7

SLPaper 2

Linda owns a field, represented by the shaded region R. The plan view of the field is shown in the following diagram, where both axes represent distance and are measured in metres. The segments [AB], [CD] and [AD] respectively represent the western, eastern and southern boundaries of the field. The function, f(x), models the northern boundary of the field between points B and C and is given by f(x) = -x^2/50 + 2x + 30, for 0 ≤ x ≤ 70

Find f'(x).

[1]

Hence find the coordinates of the point on the field that is furthest north.

[5]

Write down the integral which can be used to find the area of the shaded region R.

[1]

Find the area of Linda's field.

[4]

Calculate the percentage error in Linda's estimate.

[1]

Suggest how Linda might be able to reduce the error whilst still using the trapezoidal rule.

[3]

Find the x-coordinate of point E for the largest area of the square foundation of building EFGH.

[1]

Find the largest area of the foundation.

[5]

Question 8

SLPaper 1

Olava’s Pizza Company supplies and delivers large cheese pizzas.

The total cost to the customer, $C$ , in Papua New Guinean Kina ( $PGK$ ), is modelled bythe function

$C (n) = 34.50 n + 8.50, n \geq 2, n \in ℤ,$

where $n$ , is the number of large cheese pizzas ordered. This total cost includes a fixedcost for delivery.

State, in the context of the question,what the value of $34.50$ represents.

[1]

State, in the context of the question,what the value of $8.50$ represents.

[1]

Write down the minimum number of pizzas that can be ordered.

[1]

Kaelani has $450 PGK$ .

Find the maximum number of large cheese pizzas that Kaelani can order fromOlava’s Pizza Company.

[3]

Question 9

SLPaper 2

In memory of the legendary musician Alex Star, tribute concerts continue to attract fans worldwide, decades after his passing in 1985. The number of attendees at these concerts, ${N(t)}$ , can be modeled by

${N(t) = 200 \cdot 1.28^t}$

where ${t}$ is the number of years since 1985.

Calculate the time taken for the number of attendees to reach 100,000.

The number of attendees at a festival can be modelled by the function:

A(t) = \frac{200,000}{1 + e^{-0.4(t-10)}}

where $A(t)$ is the number of attendees and $t$ is the time in hours since the festival opened.

[2]

Write down the number of attendees in the year 1985

[2]

Calculate the number of attendees when ${t = 50}$

[2]

If the global population in 2060 is projected to be 10 billion, explain why this model for the number of attendees might be unrealistic.

[3]

Question 10

HLPaper 1

Two schools are represented by points A(2,20) and B(14,24) on the graph below. A road, represented by the line R with equation -x+y=4, passes near the schools. An architect is asked to determine the location of a new bus stop on the road such that it is the same distance from the two schools.

Find the equation of the perpendicular bisector of [AB]. Give your equation in the form y=mx+c.

[5]

Determine the coordinates of the point on R where the bus stop should be located.

[2]

Temperature

Time

Maximum

41.3^{\circ} \mathrm{C}

Minimum

0.9^{\circ} \mathrm{C}

2:00 am