IB Maths AI SL · Unit 2: Functions
IB Maths AI SL Exponential Modelling Questions
Exam-style IB Maths AI SL exponential modelling questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
- 12 questions
- Paper 1: 1
- Paper 2: 11
- 4 medium
- 6 hard
- 2 very hard
- 3 worked examples
Practise Exponential Modelling questions →
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What you need to know
Bacteria doubling every hour, a virus spreading through a population, radioactive decay — SL AI's exponential models are always applied. Learn to fit y = a·b^x from two data points. Exponential and logistic growth models overview →
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 →
Exponential Modelling worked examples
Worked example 1: Evaluating an exponential growth model · easy
The rabbit population is modelled by $P(t) = 150(1.18)^t$, $t$ in months. Calculate the expected population after $8$ months.
1. Substitute $t = 8$: $P(8) = 150(1.18)^8$.
2. Evaluate the power: $1.18^8 = 3.7588\ldots$
3. Multiply: $150 \times 3.7588\ldots = 563.83\ldots$
4. Recognise populations must be whole numbers.
5. State: $\mathbf{564}$ rabbits.
Examiner tip: Populations of living things (animals, bacteria, people) must be rounded to the nearest integer. Fractional organisms don't exist.
Worked example 2: Half-life of a depreciating asset · medium
A stamp depreciates: $V(t) = 18\,000(0.92)^t$, $t$ in years. Using your GDC, find how many years for the stamp's value to halve.
1. Target: half of $\text{€}18\,000 = \text{€}9000$.
2. Set up: $9000 = 18\,000(0.92)^t$.
3. Simplify: $0.5 = 0.92^t$.
4. Solve on the GDC (equation solver or intersection of $Y_1 = 0.92^x$ with $Y_2 = 0.5$).
5. Read: $t = 8.3129\ldots \implies \mathbf{8.31\text{ years}}$.
Examiner tip: Half-life of $y = ab^t$ is INDEPENDENT of the starting amount $a$. Just solve $0.5 = b^t$ each time.
Worked example 3: Cooling model and horizontal asymptote · hard
An ingot cools according to $T(t) = 25 + 850e^{-0.15t}$ ($t$ in minutes). Find the initial temperature and state the equation of the horizontal asymptote, explaining its meaning.
1. Substitute $t = 0$: $T(0) = 25 + 850e^{0}$.
2. Evaluate: $e^0 = 1$, so $T(0) = 25 + 850 = \mathbf{875^\circ\text{C}}$.
3. Analyse $t \to \infty$: $e^{-0.15t} \to 0$.
4. Asymptote: $T = 25 + 0 \implies \mathbf{T = 25}$.
5. Interpret: $25^\circ\text{C}$ is the ambient room temperature the ingot eventually cools to.
Examiner tip: In cooling models $y = c + ae^{-kt}$, the constant $c$ is the ambient temperature and mathematically acts as the horizontal asymptote.
Try these IB Maths AI SL exponential modelling questions
Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area.
Question 1 · medium · 5 marks · Paper 1
A heated liquid is placed in a room. Its temperature, \(T\) in \(^{\circ}\text{C}\), after \(t\) minutes is modelled by the function \(T(t) = 20 + 75e^{-0.08t}\) for \(t \ge 0\).
- Write down the initial temperature of the liquid.
- Calculate the temperature of the liquid after exactly 10 minutes.
- Write down the equation of the horizontal asymptote of the graph of \(T(t)\) and explain what it represents in the context of the problem.
Question 2 · hard · 4 marks · Paper 2
The amount of a particular medicine in a patient's bloodstream, \(M\) in milligrams, \(t\) hours after administration is modelled by \(M(t) = 250(0.85)^t\).
- Write down the initial amount of medicine administered.
- Calculate the exact half-life of the medicine in the bloodstream.
Question 3 · very hard · 8 marks · Paper 2
A heated metal object is placed in a cooling bath. Its temperature, \(T\) in \(^{\circ}\text{C}\), after \(t\) minutes is modelled by \(T(t) = 18 + 105e^{-0.12t}\).
- Write down the ambient temperature of the cooling bath.
- Find the exact time it takes for the object to reach \(40^{\circ}\text{C}\).
- A second object's temperature is modelled by \(S(t) = 15 + 120e^{-0.15t}\). Using your GDC, find the exact time when both objects are at the identical temperature.
All 12 exponential modelling questions with mark schemes →
FAQ
How many IB Maths AI SL exponential modelling questions are there?
There are 12 exam-style exponential modelling questions in the AI SL question bank (Paper 1: 1 · Paper 2: 11), graded 2 very hard, 6 hard, 4 medium. Every question has a full IB-style mark scheme (M, A and R marks).
Is exponential modelling on Paper 1 or Paper 2?
Both. In the bank, Paper 1: 1 · Paper 2: 11. 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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