2.12.1 Modelling with Functions

Modelling with functions is a powerful mathematical technique used to represent real-world situations using mathematical expressions. Functions can describe relationships between variables, predict outcomes, and solve practical problems. Here's how to approach modelling with functions:

1. Understanding the Problem

Before you can model a situation with a function, it's important to understand the scenario and identify the variables involved. Determine which variable depends on the other(s) (this will be your dependent variable) and which is the independent variable.

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Example: A company's profit $\ P$ depends on the number of units $\ x$ it sells. Here, $\ P$ is the dependent variable, and $\ x$ is the independent variable.

2. Choosing the Appropriate Function

The next step is to choose a mathematical function that best represents the relationship between the variables. Some common types of functions used in modelling include:

Linear Functions $( y = mx + c )$ : Used when the relationship is directly proportional.
Quadratic Functions $( y = ax^2 + bx + c )$ : Suitable for situations involving acceleration or areas, like projectile motion.
Exponential Functions $( \ y = ab^x )$ : Used for growth and decay, such as population growth or radioactive decay.
Logarithmic Functions ( $\ y = a \log(x) + b$ ): Often used in situations with diminishing returns, like pH levels in chemistry.

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Example: The company's profit might be a linear function of the units sold: $P(x) = 5x - 1000 ,$ where $5x$ represents the revenue and $\ -1000$ represents fixed costs.

3. Constructing the Function

Using information from the problem, construct the function by defining the parameters. This step often requires understanding how different factors influence the situation.

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Example: If it costs £2 per unit to produce an item and the company sells each unit for £5, with fixed costs of £1000, the profit function can be constructed as: $P(x) = 5x - (2x + 1000)$ Simplifying, we get: $P(x) = 3x - 1000$

4. Using the Model to Solve Problems

Once you have the function, you can use it to solve problems, make predictions, or analyse the scenario.

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Example: To find out how many units the company needs to sell to break even (i.e., when profit $\ P(x) = 0$ ): $3x - 1000 = 0$ Solve for $\ x$ : $x = \frac{1000}{3} \approx 333.33$ So, the company needs to sell approximately 334 units to break even.

5. Interpreting the Results

Finally, interpret the results in the context of the original problem. This step involves considering whether the results are realistic and meaningful.

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Example: If the company's production capacity is 500 units per month, selling 334 units is achievable and will cover the costs, leading to profit beyond this point.

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Practice Problem:

A garden hose fills a swimming pool at a rate of 10 litres per minute. The pool's volume is 2400 litres. Model the time $\ t$ (in minutes) it takes to fill the pool with a function and determine how long it takes to fill the pool to 75% of its capacity.

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Solution:

The relationship is linear: $\ V(t) = 10t .$
To find the time to fill 75% of the pool: $0.75 \times 2400 = 1800 \text{ liters}$ $10t = 1800 \quad \Rightarrow \quad t = \frac{1800}{10} = 180 \text{ minutes}$

Modelling with Functions Simplified Revision Notes for A-Level OCR Maths Pure

2.12.1 Modelling with Functions

1. Understanding the Problem

2. Choosing the Appropriate Function

3. Constructing the Function

4. Using the Model to Solve Problems

5. Interpreting the Results

Practice Problem:

500K+ Students Use These Powerful Tools to Master Modelling with Functions For their A-Level Exams.

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