A-Level · Mathematics · AQA · Mark scheme decoded
AQA A-Level Mathematics: Functions in Modelling — mark scheme explained
The short answer
In AQA A-Level Mathematics, the use of functions in modelling is a crucial skill that helps you apply mathematical concepts to real-world scenarios. This involves creating and interpreting mathematical models using functions, understanding their limitations, and refining them as necessary. ### What is Mathematical Modelling?
The question
A city's population grows exponentially. The initial population is 50,000 and the growth rate is 2% per year. Write an exponential function to model this growth and find the population after 10 years. [Paraphrased for study — not reproduced from any exam paper.]
Mark scheme, decoded
What each mark is really for — in plain English — and the wording trap that loses it.
- S1
Identify the given values: P 0 = 50,000, k = 0.02, t = 10.
- S2
Write the exponential growth function: P(t) = P 0 × e kt .
- S3
Substitute the values into the function: P(10) = 50,000 × e 0.02 × 10 .
- S4
Calculate the exponent: 0.02 × 10 = 0.2.
- S5
Evaluate the exponential term: e 0.2 ≈ 1.2214.
- S6
Multiply to find the population: P(10) = 50,000 × 1.2214 ≈ 61,070.
Model answer
Worked through, with each step tagged to the mark it earns.
- S1
Identify the given values: P 0 = 50,000, k = 0.02, t = 10.
- S2
Write the exponential growth function: P(t) = P 0 × e kt .
- S3
Substitute the values into the function: P(10) = 50,000 × e 0.02 × 10 .
- S4
Calculate the exponent: 0.02 × 10 = 0.2.
- S5
Evaluate the exponential term: e 0.2 ≈ 1.2214.
- S6
Multiply to find the population: P(10) = 50,000 × 1.2214 ≈ 61,070.
Final answer: The population after 10 years is approximately 61,070.
Common mistakes
- Using an inappropriate function for the problem — Identify the nature of the problem and select the appropriate type of function (e.g., linear, quadratic, exponential).
- Ignoring domain restrictions — Always consider the domain of the function and ensure it is appropriate for the context of the problem.
- Failing to validate the model — Compare the model's predictions with actual data and refine the model if necessary.
- Overlooking simplifying assumptions — Re-evaluate the assumptions and consider more realistic factors if necessary.
- Incorrectly interpreting results — Ensure that the interpretation makes sense in the real world and is consistent with the data and assumptions.
- Using incorrect parameters — Use the most accurate and up-to-date data available to determine the parameters of the model.
- Failing to refine the model — Continuously refine the model by incorporating new data, re-evaluating assumptions, and considering additional factors.
Where the marks go
- Full worked solution (all marking points)5 marks