Light intensity $I$ varies inversely as the square of distance $d$. When $d = 4\text{ m}$, $I = 15\text{ units}$. Find the distance at which the intensity drops to $2.4\text{ units}$.

1. Model : $I = \frac{k}{d^2}$. 2. Substitute $(d = 4, I = 15)$: $15 = \frac{k}{16}$. 3. Solve for $k$: $k = 240$. Model: $I = \frac{240}{d^2}$. 4. Substitute $I = 2.4$: $2.4 = \frac{240}{d^2} \implies d^2 = 100$. 5. Take the positive root: $d = \mathbf{10\text{ m}}$.

The force of attraction, $F$ Newtons, between two magnets varies inversely with the square of the distance, $d$ cm, between them. When $d = 4$, $F = 150$. Find the equation linking $F$ and $d$. Calculate the force when the distance is increased to $10 \text{ cm}$.

Write $F = \frac{k}{d^2}$ and substitute $d = 4,\ F = 150$: $150 = \frac{k}{16}$ M1 $k = 150 \times 16 = 2400$, so $F = \frac{2400}{d^2}$ A1 Substitute $d = 10$: $F = \frac{2400}{10^2} = \frac{2400}{100}$ M1 $F = 24 \text{ N}$ A1

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IB Maths AI SL · Unit 2: Functions

IB Maths AI SL Direct/Inverse Variation and Cubic Models Questions

Updated October 2026

Exam-style IB Maths AI SL direct/inverse variation and cubic models questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step, showing the working that earns the marks — then try the practice questions and check your working against the mark scheme.

14 questions
Paper 1: 9
Paper 2: 5
3 easy
6 medium
3 hard
2 very hard
3 worked examples

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Slope, intercept, and interpreting the gradient in context. SL AI dresses linear models as taxi fares, phone plans, and rental costs — always ask what the gradient MEANS in the real-world context. Linear functions and modelling overview →

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What's examined in AI SL direct/inverse variation and cubic models

The question bank covers these direct/inverse variation and cubic models question types (number of questions in brackets):

Direct/Inverse Variation (9)
Properties of Cubic Functions (3)
Constructing Cubic Models (2)

Direct/Inverse Variation and Cubic Models worked examples

Worked example 1: Direct variation with cubes · easy

$y$ varies directly as the cube of $x$ . When $x = 2$ , $y = 40$ . Find the constant of proportionality $k$ , and hence $y$ when $x = 3$ .

Solution

5 steps · 1 to fill in

Examiner tip: Direct variation always has the form

y = k x^{n}

; inverse variation always

y = \frac{k}{x^{n}}

. Identify which one first, THEN substitute your initial condition.

Worked example 2: Inverse square variation — light intensity · medium

Light intensity $I$ varies inversely as the square of distance $d$ . When $d = 4 m$ , $I = 15 units$ . Find the distance at which the intensity drops to $2.4 units$ .

Solution

5 steps

Examiner tip: When the problem says "inverse variation with the square (or cube) of", the exponent on the denominator must match. Forgetting the exponent is the classic error here.

Worked example 3: Maximising a cubic volume model · hard

An open box is made from a $30 cm \times 30 cm$ square by cutting squares of side $x cm$ from each corner and folding the sides up. Its volume is $V (x) = x (30 - 2 x)^{2}$ . Using your GDC, find the $x$ that maximises the volume and the maximum volume.

Solution

5 steps

Examiner tip: In real-world modelling, always set the practical domain (here

0 < x < 15

) before running the GDC max tool — otherwise you might accept a mathematical maximum that has no physical meaning.

Try these IB Maths AI SL direct/inverse variation and cubic models questions

Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area. One of them has its full worked solution here, open to everyone.

Question 1 · easy · 3 marks · Paper 1

The variable $y$ varies directly with the cube of $x$ . When $x = 3$ , $y = 54$ .

Find the constant of proportionality, $k$ , and write down the equation linking $y$ and $x$ .
Calculate the value of $y$ when $x = 5$ .

Attempt it and see the mark scheme →

Question 2 · medium · 4 marks · Paper 1

The force of attraction, $F$ Newtons, between two magnets varies inversely with the square of the distance, $d$ cm, between them. When $d = 4$ , $F = 150$ .

Find the equation linking $F$ and $d$ .
Calculate the force when the distance is increased to $10 cm$ .

Full worked solution Free, no sign-in

Write $F = \frac{k}{d^{2}}$ and substitute $d = 4, F = 150$ : $150 = \frac{k}{16}$ M1
$k = 150 \times 16 = 2400$ , so $F = \frac{2400}{d^{2}}$ A1
Substitute $d = 10$ : $F = \frac{2400}{10^{2}} = \frac{2400}{100}$ M1
$F = 24 N$ A1

Every question here has a full worked solution. See them all →

Question 3 · hard · 6 marks · Paper 2

A rectangular sheet of metal measures $30 cm$ by $20 cm$ . Four identical squares of side length $x cm$ are cut from the corners. The sides are then folded up to form an open rectangular box.

Show that the volume of the box is modelled by the cubic function $V (x) = 4 x^{3} - 100 x^{2} + 600 x$ .
State a suitable domain for $x$ in this context.
Using your Graphic Display Calculator, find the value of $x$ that maximizes the volume, and state the maximum volume.

Attempt it and see the mark scheme →

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FAQ

How many IB Maths AI SL direct/inverse variation and cubic models questions are there?

There are 14 exam-style direct/inverse variation and cubic models questions in the AI SL question bank (Paper 1: 9 · Paper 2: 5), graded 3 easy, 6 medium, 3 hard, 2 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is direct/inverse variation and cubic models on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 9 · Paper 2: 5. Practise with your GDC — AI papers expect calculator methods throughout.

Where can I get the mark schemes?

Open the AI SL Unit 2 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.

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IB Maths AI SL Direct/Inverse Variation and Cubic Models Questions

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What you need to know

What's examined in AI SL direct/inverse variation and cubic models

Direct/Inverse Variation and Cubic Models worked examples

Worked example 1: Direct variation with cubes · easy

Solution

Worked example 2: Inverse square variation — light intensity · medium

Solution

Worked example 3: Maximising a cubic volume model · hard

Solution

Try these IB Maths AI SL direct/inverse variation and cubic models questions

Question 1 · easy · 3 marks · Paper 1

Question 2 · medium · 4 marks · Paper 1

Full worked solution Free, no sign-in

Question 3 · hard · 6 marks · Paper 2

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