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SL 2.6—Modelling skills - IB Questionbank | RevisionDojo

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 1

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 ( ${ }^{\circ} \mathrm{C}$ )	88	58	43	35.5	k

Write down the decrease in temperature of the tea

during the first minute

[1]

during the second minute

[1]

during the third minute.

[1]

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

[2]

Question 7

SLPaper 2

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 :

the maximal area of the rectangle.

[4]

the side lengths for the maximal area.

[2]

the total length of the fence.

[2]

Question 8

SLPaper 2

Matt throws a discus in a competition, and its flight can be modelled by the function $y=\frac{x(22-x)}{20}+1.8,0 \leq x \leq 23.5$ 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]

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

[2]

Question 9

SLPaper 2

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.

[4]

Using the value for the area from part 1.,calculate the side lengths and the total length $L$ of the fence.

[2]

Question 10

SLPaper 1

Jack is a contracted carpenter who earns an annual salary of $\$ 55000 $. Additionally, Jack sells innovative wooden art at a market. He goes to the market twice a month and he generally earns$ $ 450$ from the market.

Estimate Jack's total annual income.

[2]

Jack's actual total annual income over the year is $\$ 68000$.

Calculate the percentage error between your answer in part 1 and Jack's actual total annual income.

[2]

SL 2.6—Modelling skills Questionbank