IA idea · Calculus & optimisation
The lifeguard problem: what is the fastest route to a swimmer?
Research question
Where should a lifeguard enter the water to reach a swimmer in the least time, given measured running and swimming speeds, and does the optimal path obey Snell's law?
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Why it makes a good exploration
Running is faster than swimming, so the straight line is not the fastest route. The calculus answer turns out to be the same law that bends light at water — a beautiful link you can test with your own speeds.
The mathematics you'll need
- Distance with Pythagoras
- Time as a function of the entry point
- Differentiation to minimise time
- Showing the optimum satisfies sin θ₁/v₁ = sin θ₂/v₂
Where the data comes from
Time yourself running 20 m on sand (or grass) and swimming 20 m; set a realistic beach geometry.
- Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.
A possible outline
- Define the geometry and variables.
- Write total time T(x) and differentiate.
- Solve for the optimal entry point, numerically if necessary.
- Show the condition is Snell's law.
- Compare with the straight-line path and the “run as far as possible” path.
Pitfalls that cost marks
- Using speeds you have not measured or sourced.
- Solving numerically without checking it is a minimum.
- Ignoring waves and currents without comment.
Showing personal engagement
- Use your own running and swimming speeds.
- Test whether a dog playing fetch approximates the optimum (a famous study).
- Discuss what a real lifeguard would do.
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 | Distance with Pythagoras; Time as a function of the entry point |
| AA HL | Good fit | Distance with Pythagoras; Time as a function of the entry point |
| AI SL | Good fit | Distance with Pythagoras; Time as a function of the entry point |
| AI HL | Good fit | Distance with Pythagoras; Time as a function of the entry point |
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
Reflection (D)
For this idea, start with the weak points: using speeds you have not measured or sourced; and solving numerically without checking it is a minimum. Say how each one affects your answer.
Use of mathematics (E)
SL: distance with Pythagoras; time as a function of the entry point; differentiation to minimise time; showing the optimum satisfies sin θ₁/v₁ = sin θ₂/v₂, each explained in your own words and carried out correctly.
How calculus explorations reach the top bands in general: Calculus & optimisation IA ideas. Guides: A · B · C · D · E · self-grade sheet.
Taking it further
Add a current in the water and re-optimise.
Extending it for HL
Add an HL technique the question genuinely needs: what an HL calculus exploration adds.
See a complete IA, marked
How long does a skydiver take to reach terminal velocity? (AA HL) is a complete calculus exploration, marked on every criterion.
Frequently asked questions
Which IB Maths course is this IA idea for?
It works for AA SL, AA HL, AI SL and AI HL. We rate it Solid: it needs some independent work beyond class examples.
What mathematics does this IA use?
Distance with Pythagoras, time as a function of the entry point, differentiation to minimise time and showing the optimum satisfies sin θ₁/v₁ = sin θ₂/v₂.
Where can I get data for this IA?
Time yourself running 20 m on sand (or grass) and swimming 20 m; set a realistic beach geometry. Checked sources and tools: Desmos graphing calculator.
Turn this idea into your IA
The planner opens with “The lifeguard problem: what is the fastest route to a swimmer?” filled in: the question, the maths and the data sources.
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