IB Mathematics AI SL Modelling MAI Study Notes - New Syllabus

IB Mathematics AI SL Modelling MAI Study Notes

LEARNING OBJECTIVE

Use the modelling process described in the “mathematical modelling” section to create, fit and use the theoretical models

Key Concepts:

Strategy for Modelling Functions

MAI HL and SL Notes – All topics

Mathematical modelling involves the following steps:

1. Identifying the Problem

Understand the real-world situation to be modelled.

2. Making Assumptions and Defining Variables

Simplify the problem by making reasonable assumptions and define variables clearly.

3. Formulating the Model

Develop mathematical relationships (equations or functions) that represent the situation.

4. Solving the Model

Use appropriate mathematical methods to find solutions.

5. Interpreting the Solution

Translate mathematical results back into the context of the original problem.

6. Validating the Model

Compare model predictions with real data to assess accuracy.

7. Refining the Model

Adjust the model as necessary to improve its fit or applicability.

Example:

Alfred joins a film streaming website. There are two options:

Per-film: First film costs \$1, then \$2.50 for every additional film.
Subscription: \$5.50 to join, then \$3 per month.

(a) Create a linear model for the per-film option.

(b) If he watches two films per month, after how many months does the subscription become better?

▶️Answer/Explanation

Solution:

(a) Let $ f $ be the number of films watched.

The first film costs \$1, and the rest cost \$2.50 each:
$ C = 1 + 2.5(f – 1) = 2.5f – 1.5 \quad \text{[for } f \ge 1 \text{]} $

(b) If Alfred watches 2 films per month, then $ f = 2m $, where $ m $ is the number of months.

Per-film cost model:
$ C = 2.5(2m) – 1.5 = 5m – 1.5 $

Subscription model:
$ C = 5.5 + 3m $

Set the two models equal:
$ 5m – 1.5 = 5.5 + 3m $
$ 2m = 7 \Rightarrow m = 3.5 $

∴ Starting from month 4, the subscription becomes cheaper.

Types of Models

Common functions used in modelling include:

Linear Models: Represent constant rate of change.
Quadratic Models: Model situations with acceleration or deceleration.
Exponential Models: Describe rapid growth or decay processes.
Trigonometric Models: Used for periodic phenomena.

Selecting the appropriate model depends on the nature of the data and the context of the problem.

Example:

Uganda had a population of $44.5$ million on $1/1/19$ and has a growth rate of approximately $4.9\%$ per year.

(a) Develop a model for Uganda’s population over time.

(b) Estimate the population of Uganda on $1/1/2028.$

(c) The Ugandan government has stated that due to its small area, they don’t think they could sustain a population of more than $95$ million people.
At the current pace, what month would they pass this figure?

(d) Japan had $126.5$ million people in $2019$, but its population is shrinking at $2.1\%$ per annum.
When should Uganda’s population overtake Japan’s?

▶️Answer/Explanation

Solution:

(a) Let $ t $ be the number of years since 1/1/2019. Population model:

$ P = 44.5(1.049)^t $

(b) For 1/1/2028, $ t = 9 $:
$ P = 44.5(1.049)^9 \approx \boxed{68.4 \text{ million}} $

(c) Find when population exceeds 95 million:

$ 44.5(1.049)^t = 95 $
$ \Rightarrow 1.049^t = \frac{95}{44.5} \approx 2.13483 $
$ \Rightarrow t = \log_{1.049}(2.13483) \approx 15.85 $

≈ 15 years and 10 months → November 2034

(d) Japan’s model: $ P = 126.5(0.979)^t $

Set populations equal:
$ 44.5(1.049)^t = 126.5(0.979)^t $
Solve graphically or numerically:

$ t \approx 15.13 \Rightarrow \boxed{\text{February 2034}} $

Using Technology

Graphing calculators or software can assist in:

Plotting data points
Fitting curves to data
Calculating regression equations
Visualizing the model’s behavior

Technology aids in refining models and assessing their fit to real-world data.

Evaluating Models

When assessing a model:

Appropriateness: Does the model suit the context?
Accuracy: How well does the model fit the data?
Limitations: What are the constraints or assumptions of the model?
Predictive Power: Can the model reliably predict future outcomes?

Example

Scenario:

A hot air balloon ascends vertically, covering 450 meters in the first minute. Each subsequent minute, it travels 82% of the distance covered in the previous minute.

Model: This situation can be modelled using a geometric sequence:

▶️Answer/Explanation

Solution:

$
d_n = 450 \times (0.82)^{n – 1}
$

Where $ d_n $ is the distance covered in the $ n $th minute.

Total distance in 10 minutes:

$
S_{10} = 450 \times \frac{1 – (0.82)^{10}}{1 – 0.82}
$

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