Maths IA · Calculus · topic 21 of 97

The lifeguard problem: where should you enter the water? · IB Maths IA idea: guide, sample graph and step-by-step plan

You can run faster than you can swim, so the straight line to the swimmer is not the quickest way. Somewhere along the shore there is a best place to enter the water, and the maths for it also describes how light bends.

• ✓ you like optimisation problems with a real person in them
• ✓ you want to run, swim or walk a route and time it yourself
• ✓ you are curious why light bends when it enters water

The big idea

Moving through two different surfaces at two different speeds makes the total time a function of one number: where you cross the boundary. Write the time as two square roots, one for each leg, divide each by its speed and differentiate. Setting the derivative to zero gives an equation you can solve numerically, and when you tidy it, it says that the sine of each angle divided by its speed is the same. That is Snell's law, and the same reasoning is behind Fermat's principle of least time for light. The calculus is the easy part. The marks come from your measured speeds: how you get them, how uncertain they are, and how much the best route moves when they change.

Make it yours: pick one twist

Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.

Swipe for more twists →

Your Maths IA research question: run the check

Write your own question first. Then make sure it ticks every box:

• Names the places, the distances a, b and d, and what the two speeds are.
• Says how the speeds will be measured, with how many trials, and how uncertainty will be handled.
• Says what will be solved or compared: the best entry point, the time saved or the link to Snell's law.
• Includes a test of the result, such as timing the real routes or measuring the angles.
• Is narrow enough for 12 to 20 pages and goes beyond solving one textbook problem once.

The model

Run on sand, swim in water, and choose where to enter. Mark your own distances and speeds, and the angles at the entry point.

How your Maths IA could be laid out

The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.

What your Maths IA graph could look like

Made-up numbers to show the form of a strong graph, not the answer: total time against the entry point, with computed values. Yours will look different. example only

• Total time against the entry point, so the minimum can be read and the flatness of it seen.
• Axis labels give units: distance in metres, time in seconds.
• Mark the calculated minimum, then show timed routes as points to test it.
• Add error bars from your repeated timings so the reader can judge the agreement.
• Another graph to consider: the best entry point against the ratio of the two speeds.

The maths in action

The graph above was a sketch with made-up numbers. This figure works the idea through from the model itself, so you can see what a finished figure looks like. The source line under it says where every number comes from.

How much time does the best route save?

Rank four routes on one number: total time. Work it out for your own geometry and explain why the best route is not the shortest.

Your game plan

Tick steps off as you go. We'll remember on this device.

0 of 10 done

1. Read and predictFind out what the lifeguard problem asks. Before any calculation, sketch the route you think is fastest, and say why.
2. Choose the placePick a real layout: a beach, a field and a pond, or a lane and a road. Measure the distances a, b and d.
3. Measure the speedsTime yourself over a measured distance on each surface, several times. Record the mean and the spread.
4. Trial run (1 lesson)Time two or three actual routes over your layout. Is the timing method good enough to tell the routes apart? Fix it now.
5. Build T(x)Use Pythagoras for each leg and divide by the speed. Graph T(x) and read an approximate minimum before differentiating.
6. Differentiate and solveFind T'(x), set it to zero and solve, by hand or numerically. Show the minimum is a minimum, and compare with your graph.
7. Link to Snell's lawRewrite the condition with the sines of the two angles. Check it numerically with your own solution.
8. Go further, with a reasonTest the sensitivity to the speeds, add a third surface or curve the shore. Say what question the extra step answers. HL: this is where the rigour for criterion E comes from.
9. Reflect as you goAfter every result, write one line on what it means for a rescuer and what it assumes. These lines become your evaluation.
10. Cut, check, then sendCheck units everywhere, every graph labelled, every symbol defined once and the conclusion answers your aim. Then send your teacher one draft.

The maths you'll use

• Pythagoras and modelling – Write each leg as or and divide by its speed to form .
• Differentiation – Use the chain rule on both square roots to find and set it to zero.
• Optimisation – Show at the solution, and use a graph and a table of values to confirm the minimum is inside the shore.
• Trigonometry – Replace by to turn into Snell's law.
• Numerical solving – Solve on a calculator or spreadsheet when it cannot be done by hand, and test the solution against the model.
• Uncertainty – Propagate the uncertainty in the measured speeds to the best entry point and the minimum time.

Key equations

Syllabus links

• 3.2Right-angled trigonometry, sine and cosine rules
• 5.3Differentiating powers of x
• 5.6Standard derivatives; chain, product and quotient rules
• 5.8Maxima, minima, inflexion and optimisation
• 3.8Solving trigonometric equations

• 3.2Right-angled trigonometry, sine and cosine rules
• 5.3Differentiating powers of x
• 5.6Stationary points
• 5.7Optimisation in context
• 5.9Further derivatives; chain, product, quotient rules; related ratesHL

Read up first

• Differentiation (AA HL) →
• Trigonometry (AA HL) →
• Differentiation (AI HL) →

Data and sources to start from

Time yourself running and swimming (or walking on two surfaces) over measured distances, several trials each, and time routes through your chosen entry points.

References to cite

1. Stewart, J. (2015). Calculus: Early transcendentals (8th ed.). Cengage Learning.
2. Feynman, R. P., Leighton, R. B., & Sands, M. (2011). The Feynman lectures on physics (New millennium ed., Vol. 1). Basic Books. (Chapter 26, optics: the principle of least time.)

Cite what you actually use, in your own format, and say where each number came from.

Traps to avoid in this Maths IA

• Assuming the straight line is fastest because it is shortest. The speeds are different.
• Mixing up which leg gets which speed, so the time function is wrong.
• Setting T'(x) = 0 but never checking that it is a minimum, or that it lies inside the shore.
• Using angles measured from the shoreline in Snell's law instead of from the normal to it.
• Quoting speeds with no measurements and no uncertainty, then reporting the best point to three decimals.
• Treating Snell's law as a coincidence, rather than showing it comes from the equation you solved.

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