Maths IA · Calculus · topic 15 of 40

Draining tank: predict the time to empty · IB Maths IA idea: guide, sample graph and step-by-step plan

stretching, but very rewarding

Water leaves a tank fast at first and slowly at the end. Can a differential equation tell you exactly when it runs dry?

Difficulty●●● Stretch
KitA funnel, cone or bottle, a small hole, a ruler, a phone for timing
TimeAbout 14 hours
FitsAA HL and AI HL
  • ✓ you like a problem where maths and a real object meet
  • ✓ you can film a container draining
  • ✓ you are happy with some physics assumptions to check

The big idea

the physics assumptions are marks too

The rate at which the depth falls depends on how fast water leaves the hole and how wide the surface is. Both change with depth, so you get a differential equation in the depth. If the variables separate, you can solve it, find the depth at any time and the time to empty. Then you drain a real container and film it. The physics behind the outflow rate has assumptions, so half the marks are in deciding which to trust and how to test them. At HL, show the solution method and justify each step.

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 →

Change the shape

Cylinder, cone, hemispherical bowl or a bottle that you approximate by a solid of revolution.

→ The surface area as a function of depth changes, and so does the differential equation. Some shapes separate neatly, others need a numerical method.
Change the hole

Try holes of different sizes and shapes, and compare the time to empty.

→ A parameter in your model that you can estimate from data, and test: does the fitted value stay the same for all holes?
Fit the outflow constant

Treat the loss at the hole as an unknown constant and fit it to the depth-time data.

→ You are using the model to learn a number, then checking whether that number is believable and consistent.
Solve it numerically too

Use Euler's method or a spreadsheet to solve the equation without the closed form.

→ You compare exact and numerical answers, discuss the step size and show the numerical method converges.
Design the container

Ask what shape would make the depth fall at a constant rate. This is how an ancient water clock worked.

→ You solve an inverse problem: given the behaviour you want, find the shape. A genuine HL-level extension.
Test the assumptions

Check for a long thin outlet, a hole high on the side, or a lid that stops air entering.

→ Evaluation points with evidence, not guesses. You can model the effect or show it is small.

Your Maths IA research question: run the check

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

  • Names the container shape and the quantity you will predict (the time to empty, or depth at a given time).
  • Says what stays fixed (the hole, the starting depth, the liquid) and what you vary.
  • States the physical assumption for the outflow, and how you will check it against data.
  • Says how the model will be compared with measurements, and what a 'good' agreement would be.
  • Needs a differential equation you set up and solve yourself, with working shown.

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The model

A conical tank with a small hole. Label depth, surface radius and hole area, and say what you measure and how often.

rhruler or video:read h at each timesmall hole: area aweigh what drainsarea = π r²,r depends on h

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.

Aim and context~1 page
The container you chose, why, and what you want to predict. A first guess at how depth changes with time.
The physics you use~1.5 pages
The rule for the speed of water leaving the hole, with its source and its assumptions stated plainly.
Setting up the equation~2 pages
Surface area as a function of depth, volume balance, and the differential equation, with every symbol defined.
Solving it~2.5 pages
Separation of variables, integration, the constant, and the time to empty. Each step justified. Units checked.
The experiment~2 pages
Container, hole, how you measured depth and time, repeats, with a table and uncertainties.
Model against data~2.5 pages
Predicted and measured depth against time on one graph, the fitted outflow constant, and a number for the difference.
Conclusion~1 page
Back to the aim: how well did the model predict the time to empty?
Evaluation~2 pages, the biggest part!
Which assumption is weakest? What does the data say about it? What would you change first?
Bibliographynot counted
Sources for the outflow rule and for solving the equation.

What your Maths IA graph could look like

two models: show the difference

Made-up numbers, to show the form of a strong graph, not the answer. Yours will look different. example only

020406080100120024681012141618202224Time t / s (± 0.5 s)Depth h / cm (± 0.2 cm)model with fitted outflow constantmodel with the outflow constant 1.67 times largermade-up measured depthsfalls faster near the end: why?(think area at the surface)ILLUSTRATIVE · MADE-UP DATA
  • Depth against time, with the model through the data. The reader can judge the fit at a glance.
  • A second curve shows what a different outflow constant would do, so the choice of constant is visible, not hidden.
  • Error bars on depth and on time. Note which is more important here.
  • Mark where the model predicts the tank is empty, and compare with the time you measured.
  • Another graph to consider: depth to a power against time. If the model is right, it is a straight line, and the gradient gives your outflow constant.

Your game plan

trial run: how long does it take?

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

0 of 10 done

  1. Choose the shape and predictPick a container you can approximate by simple geometry. Sketch how you think the depth will change over time.
  2. Find and test the outflow ruleFind the rule for speed at the hole, note its assumptions, and check it on the simplest case, a cylinder.
  3. Set up the equationWrite the volume balance, express the area in terms of depth, and write the differential equation.
  4. Solve it by handSeparate the variables, integrate, use the initial condition and find the time to empty. Check units and a limiting case.
  5. Trial run (1 lesson)Drain the container once with a stopwatch or camera. Does the flow stay smooth? Can you read the depth easily?
  6. Collect dataFilm the draining against a ruler, repeat at least three times, and extract depth against time at regular steps.
  7. Fit and compareFit the unknown constant to your data. Is it the same for each repeat? Plot model and data together.
  8. Check one assumptionPick the assumption you trust least and test it with a new run or a numerical model.
  9. Reflect as you goAfter each result, ask what it tells you about the physics. HL: say what you justified and what you only assumed.
  10. Cut, check, then sendCheck every step of the solution reads as proper notation and nothing jumps. Give your teacher one draft.

Maths you'll need

Traps to avoid in this Maths IA

  • Treating the physics as given: the outflow rule has conditions. Say what they are and whether you met them.
  • Dropping the constant of integration, or finding it with the wrong starting condition.
  • Depth measured from the wrong place, so the model and the data use different zeros.
  • A tank so large or a hole so small that the experiment takes an hour. Test the scale in the trial run.
  • Taking a time to empty from a graph of the last few drops, where the flow is not smooth.
  • HL: an equation lifted from a website, with no sign you derived it.
  • Claiming good agreement without a number for the difference.

More Maths IA ideas like this

Make sure it's yours

This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.

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