Updated 12 October 2026 · By Pete Bromfield, former Head of Maths with IB examining experience

IA idea · Differential equations & dynamics

How long would a lake take to flush out a pollutant?

AA HLAI HL Solid Also in: Environment

Research question

Using a mixing differential equation with a lake's published volume and outflow, how long would it take for a pollutant's concentration to fall to 5% of its initial level, and how sensitive is that to the well-mixed assumption?

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Why it makes a good exploration

Environmental agencies use exactly this model to estimate recovery times after spills. Applying it to a real lake links a standard HL differential equation to a real decision.

The mathematics you'll need

Rate in − rate out modelling
First-order linear DE and its exponential solution
Half-life and time to 5%
Comparing continuous and discrete (Euler) solutions

Modelling the data, step by step

This idea usually needs an exponential model. See it worked step by step, with a criterion tip at every step: Exponential y = a e^(kx) + c and y = a·bˣ.

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

Find a lake's volume and outflow rate from a published source (national water agency or encyclopaedia) and cite it.

OpenStreetMap — Free map with exact coordinates of schools, hospitals, shops and stations; export or read off coordinates.

Real data for this idea

Primary source: OpenStreetMap. Read off coordinates (right-click, “Show address”) or use Export for a small area. Licence: Open Database Licence: credit “© OpenStreetMap contributors”; data you build from it must be shared alike.

OpenStreetMap is under the Open Database Licence: credit “© OpenStreetMap contributors”. Links checked October 2026; we link to the data and never copy it. More free data sets.

A possible outline

Derive dC/dt = −(Q/V)C for a clean inflow.
Solve and interpret Q/V as a flushing rate.
Apply to two lakes with different volumes.
Extend to a continuing pollutant inflow.
Reflect on stratification and incomplete mixing.

Pitfalls that cost marks

Unit errors between m³/s and km³.
Treating a huge lake as well mixed without comment.
Unsourced volume and flow figures.

Showing personal engagement

Choose a lake near you or one with a real pollution history.
Compare the model with a real recovery reported in the news.
Design a two-basin model and compare.

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

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Not a natural fit	The core technique sits in the AI course or at HL; an AA SL student could use it only as clearly explained new mathematics.
AA HL	Good fit	Rate in − rate out modelling; First-order linear DE and its exponential solution
AI SL	Not a natural fit	The mathematics is mainly AA or HL (calculus or proof beyond AI SL); an AI SL version would need a data-driven, technology-based approach.
AI HL	Good fit	Rate in − rate out modelling; First-order linear DE and its exponential solution

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: unit errors between m³/s and km³; and treating a huge lake as well mixed without comment. Say how each one affects your answer.

Use of mathematics (E)

HL: rate in − rate out modelling; first-order linear DE and its exponential solution; half-life and time to 5%; comparing continuous and discrete (Euler) solutions, each explained in your own words and carried out correctly.

How differential equations explorations reach the top bands in general: Differential equations & dynamics IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

Model a lake that receives a seasonal inflow (sinusoidal Q) and solve numerically.

Extending it for HL

Add an HL technique the question genuinely needs: what an HL differential equations exploration adds.

See a complete IA, marked

Is it cheaper to leave the heating on low overnight? A two-temperature model of my bedroom (AI HL) is a complete differential equations exploration, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AA HL and AI HL. We rate it Solid: it needs some independent work beyond class examples.

What mathematics does this IA use?

Rate in − rate out modelling, first-order linear DE and its exponential solution, half-life and time to 5% and comparing continuous and discrete (Euler) solutions.

Where can I get data for this IA?

Find a lake's volume and outflow rate from a published source (national water agency or encyclopaedia) and cite it. Checked sources and tools: OpenStreetMap.

Turn this idea into your IA

The planner opens with “How long would a lake take to flush out a pollutant?” filled in: the question, the maths and the data sources.

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