Introduction to Maths and Science in Product Design

In the world of Design and Technology, math and science aren't just subjects you study in a different classroom—they are the "engine room" of a successful product. When we talk about Managing design, cost and production, we need these skills to ensure our products are strong enough, don't cost too much to make, and fit the people they are designed for. Don't worry if you find math a bit daunting; these notes break down exactly what you need for the Edexcel A Level exam, focusing on practical applications you can use in your own projects!

1. Working with Numbers, Costs, and Waste

To manage a production line effectively, you need to be confident with numbers. This helps in calculating how much material to buy and how much a product will sell for.

Percentages and Percentiles

Percentages are most commonly used to calculate material waste. In industry, wasting material is wasting money!

The Formula for Percentage Waste:
\(\text{Percentage waste} = \left( \frac{\text{waste material}}{\text{total material}} \right) \times 100\)

Example: If you use a sheet of polymer weighing \(2kg\) but \(0.4kg\) is trimmed off as scrap, your waste is:
\(\frac{0.4}{2} \times 100 = 20\%\)

Percentiles are used in anthropometrics (human measurements). Most designers aim for the "5th to 95th percentile." This means the product is designed to fit 90% of the population, excluding only the extreme smallest 5% and largest 5%.

Ratios and Scaling

Ratios are essential when creating scale drawings or models. A scale factor helps you translate a small drawing into a full-sized prototype.

The Formula for Scale Factor:
\(\text{Scale factor} = \frac{\text{drawing dimension}}{\text{actual dimension}}\)

Example: If a chair is \(1000mm\) tall and your drawing is \(100mm\) tall, your scale is \(1:10\).

Key Takeaway:

Always keep track of your waste and use percentiles to ensure your design is inclusive for the majority of users.

2. Calculating Surface Area and Volume

Why do we need this? To know how much paint or finish is required (surface area) and how much raw material or liquid a container can hold (volume).

Important Formulas to Remember:

You should be able to recall these from GCSE Higher Tier math:

  • Area of a rectangle: \(\text{length} \times \text{width}\)
  • Area of a triangle: \(0.5 \times \text{base} \times \text{height}\)
  • Volume of a cuboid: \(\text{length} \times \text{width} \times \text{height}\)
  • Volume of a cylinder: \(\pi \times r^2 \times h\)

Quick Tip: Always check your units! If your measurements are in \(cm\), your area will be \(cm^2\) and your volume will be \(cm^3\). Manufacturers usually work in \(mm\).

3. Geometry and Trigonometry in Design

Designers use geometry to ensure parts fit together perfectly and to calculate angles for ergonomic support.

Right-Angled Trigonometry

You may need to calculate the length of a support beam or the angle of a chair back. The most common tool is Tan.

\(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\)

Example: To find the angle of a ramp that is \(200mm\) high (opposite) and has a base of \(1000mm\) (adjacent):
\(\tan(\theta) = \frac{200}{1000} = 0.2\)
\(\theta = \tan^{-1}(0.2) \approx 11.31^\circ\)

Coordinates and Datum Points

When using CAD/CAM (like a CNC router or laser cutter), the machine needs a datum point. This is the \((0,0)\) coordinate from which all other measurements are taken. Setting an accurate datum point ensures the machine cuts in the right place, reducing errors and material waste.

4. Science Skills: Degradation and Material Choice

A designer must understand the science behind why materials fail so they can prevent it. This is part of Designing for Maintenance (see Topic 9 for more on sustainability).

Conditions that Cause Degradation

Products are constantly under attack from their environment. You need to describe these factors:

  • Oxidation and Corrosion: Metals like mild steel react with oxygen and moisture to form rust. This can be prevented through finishes like galvanisation or powder coating (Topic 3.5).
  • UV Degradation: Some polymers become brittle and change colour when exposed to sunlight.
  • Biological Attack: Woods can rot or be attacked by insects if they are not treated with preservatives.
  • Chemical Attack: Exposure to solvents or acids can weaken certain materials.

Physical Properties and Uses

You must be able to explain how a material's science relates to its job. For example:

  • Conductivity: Why copper is used for wires (electrical) or aluminium for saucepans (thermal).
  • Strength vs. Weight: Why composites like CFRP (Carbon Fibre) are used in high-performance bicycles.
  • Hardness: Why high-carbon steel is used for cutting tools to keep a sharp edge.
Did you know?

Choosing the right material isn't just about strength; it's about the Product Life Cycle. Using a biodegradable material might be better for a single-use product, even if it is technically "weaker" than a polymer.

5. Data Handling and Statistics

In the "Managing Production" section, we use charts and graphs to make sense of Market Research and Quality Control.

Graphs and Charts

You should be able to analyse:

  • Bar Charts/Pie Charts: Great for seeing user preferences (e.g., "70% of users prefer the blue handle").
  • Line Graphs: Useful for tracking Product Life Cycles (Introduction, Growth, Maturity, Decline).

Probability and Likelihood

In manufacturing, we use probability to predict Quality Control issues. If a machine has a 1 in 1000 chance of making a mistake, a manufacturer can calculate the cost of "faulty goods" and decide if the process is efficient enough.

6. Summary of Formulae to Recall

Don't worry about memorising complex physics equations, but you must know these for the exam:

1. \(\text{Area of Rectangle} = L \times W\)
2. \(\text{Area of Triangle} = 0.5 \times B \times H\)
3. \(\text{Volume of Cuboid} = L \times W \times H\)
4. \(\text{Volume of Cylinder} = \pi \times r^2 \times h\)
5. \(\% \text{ Waste} = \left( \frac{\text{waste}}{\text{total}} \right) \times 100\)
6. \(\text{Total Cost} = \text{quantity} \times \text{unit cost}\)
7. \(\text{Scale Factor} = \frac{\text{drawing size}}{\text{actual size}}\)
8. \(\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}\)

Common Mistakes to Avoid

  • Mixing Units: Never calculate using \(mm\) and \(cm\) in the same sum. Convert everything to \(mm\) first!
  • Radius vs. Diameter: In volume calculations for cylinders, the formula uses the radius (\(r\)). If the exam gives you the diameter, halve it first!
  • Forgetting Working: The command word Calculate means you must show your steps. Even if the final answer is wrong, you can get marks for the correct formula.
  • Calculator Mode: Ensure your calculator is in Degrees (DEG) mode, not Radians, when doing trigonometry.
Quick Review:

Math and science in DT are tools to help you minimise waste, maximise profit, and ensure safety. Whether you are scaling a drawing or choosing a material that won't corrode, these calculations are the foundation of professional product design.