IA idea · Optimisation & linear programming

What speed minimises the total cost of a long drive?

AA SLAI SLAA HLAI HL Solid Also in: Calculus, Finance, Kinematics

Research question

If fuel use per kilometre rises with speed but driver time also costs money, what speed minimises the total cost of a 300 km journey, and how does that speed depend on fuel price and the value of time?

Adapt it: change the place, the data or the comparison until the question is yours.

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Why it makes a good exploration

Two costs pull in opposite directions, which makes a genuine minimum. Fitting your own fuel-consumption model before optimising adds a modelling step that most textbook versions skip.

The mathematics you'll need

Fitting a quadratic (or other) model to fuel consumption against speed
Total cost as a function of speed, including a time cost ∝ 1/v
Differentiation to find the minimum; second-derivative check
Sensitivity of the optimum to fuel price

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

Modelling the data, step by step

This idea usually needs a quadratic model. See it worked step by step, with a criterion tip at every step: Quadratic: three points, completing the square, regression.

Model your own data Paste it from Desmos, GeoGebra or a spreadsheet and get the same play-by-play with your numbers. New to modelling? Start with the modelling workflow. Writing it up? The IA modelling planner comments on each paragraph as you draft — it never writes it for you.

Where the data comes from

Record fuel consumption at different steady speeds from a family car's trip computer (as a passenger, on safe roads), or find published fuel-economy-by-speed figures from a reputable source and cite them.

Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

A possible outline

Collect or source consumption-against-speed data.
Fit and compare two models.
Build the total cost function and minimise it.
Explore how the optimum changes with fuel price and value of time.
Reflect on speed limits, safety and the data's reliability.

Pitfalls that cost marks

Optimising without a real consumption model.
Collecting data unsafely; use a trip computer as a passenger only.
Ignoring speed limits in the conclusion.

Showing personal engagement

Use your own family car.
Choose a journey you actually make.
Find the fuel price at which the optimum crosses the speed limit.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Fitting a quadratic (or other) model to fuel consumption against speed; Total cost as a function of speed, including a time cost ∝ 1/v
AA HL	Good fit	Fitting a quadratic (or other) model to fuel consumption against speed; Total cost as a function of speed, including a time cost ∝ 1/v
AI SL	Good fit	Fitting a quadratic (or other) model to fuel consumption against speed; Total cost as a function of speed, including a time cost ∝ 1/v
AI HL	Good fit	Fitting a quadratic (or other) model to fuel consumption against speed; Total cost as a function of speed, including a time cost ∝ 1/v

Level: Solid. Needs some independent work beyond class examples. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Personal engagement (C)

Optimise a decision that is really yours or your school's (a timetable, a budget, a delivery), gather the real constraints yourself, and say which ones you chose to ignore and why.

Reflection (D)

Compare the mathematical optimum with what people actually do, and test how sensitive the optimum is: which constraint, if relaxed a little, would change the answer most? For this idea, start with: optimising without a real consumption model — say how it affects your answer.

Use of mathematics (E)

SL: An objective function and constraints set up from the context, solved correctly (graphically for two variables, or with differentiation), the optimum checked and interpreted, and any new method such as linear programming explained in your own words.

HL: Optimisation with two or more variables, a justified numerical search, a proof that the optimum lies at a vertex, or a sensitivity analysis with calculus, used because the problem needs it.

Criteria A and B (presentation and communication) work the same way for every idea: see the guides to Criterion A and Criterion B.

Taking it further

Treat fuel price and value of time as parameters and find the optimum speed as a function of their ratio.

Extending it for HL

Add a second variable or a non-linear constraint, use a numerical search where calculus alone is not enough, and analyse how the optimum moves as a parameter changes.

See a complete IA, marked

Our annotated exemplar Is a hanging chain a parabola? Comparing catenary and quadratic models (AA SL) asks a different question, but shows how a complete optimisation exploration is structured and marked, with an examiner's comment on every criterion. Free excerpts and the full marking table are on its page.

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Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Criteria checklistCheck your plan or draft against the A–E descriptors and self-grade before you hand it in. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, an IB examiner, reads your draft and talks it through with you (€100).