A mathematical model is a deliberately simplified description of a situation. It relates quantities, supplies predictions and makes assumptions that can be checked. Its variables and assumptions must describe the situation being studied.
The task is to choose a relationship that answers a particular question at an appropriate level of detail. This lesson follows one hypothetical tank from measurement through prediction, testing and revision, rather than collecting unrelated model types.
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Begin with a question and a defined system
A rectangular tank has internal dimensions 80 cm by 50 cm by 40 cm, giving capacity 160 L. It initially contains 20 L. A pump is expected to add water at approximately 12 L/min. How long will it take to fill?
Before writing an equation, define the system. Assume the tank is level, its cross-section is constant, there is no leak or outlet flow, and the inflow remains constant until an automatic stop at capacity. These are model assumptions, not consequences of the dimensions.
The same dimensions were used in Length, Area and Volume Units. Reusing them lets the geometry and rate calculation be checked against one another.
Define variables, parameters and units
Let be elapsed time in minutes and the water volume in litres. The initial value is 20 L, the rate parameter is 12 L/min, and capacity is 160 L. The uncapped quantity equation is
When denotes its numerical value in minutes, we may abbreviate this as , with reported in litres. Keep this numerical-unit convention visible; replacing minutes with seconds would change the coefficient.
The rate is a slope, while the initial volume is an intercept. They have different units and different physical roles.
Solve and check the prediction
The volume to be added is L. Therefore,
That is 11 minutes 40 seconds, not 11 minutes 67 seconds. Substitution confirms L.
The rectangular cross-section is cm². One centimetre of depth therefore represents 4000 cm³, or 4 L. Initial depth is 5 cm and the modelled level rises at 3 cm/min. Reaching 40 cm requires another 35 cm, again taking minutes. This second representation checks both the units and the geometry.
Restrict the model to physically meaningful outputs
The uncapped line applies only for under the assumptions above. If the pump stops at capacity, a complete idealised model is
If instead the pump continues and water overflows, the stored-volume model may still cap at 160 L, but the delivered volume and overflow are separate quantities. A tank cannot store 200 L merely because an unrestricted line predicts that value.
Physical capacity changes the formula that applies
Diagram explanation and exact data
The fill time solves 20 + 12t = 160, so t = 35/3 min. Before this time volume is linear; afterwards it is constant. These two pieces describe a controlled filling process, not continued overflow.
| Time (min) | Volume (L) |
|---|---|
| 0 | 20 |
| 5 | 80 |
| 10 | 140 |
| 160 | |
| 20 | 160 |
Compare predictions with separate observations
For teaching, use the following synthetic observations. They are stipulated readings, not actual measurements taken by CTS.
| Time (min) | Observed volume (L) | Predicted volume (L) | Residual: observed minus predicted (L) |
|---|---|---|---|
| 0 | 20 | 20 | 0 |
| 2 | 44 | 44 | 0 |
| 5 | 79 | 80 | −1 |
| 8 | 116 | 116 | 0 |
| 10, held out | 136 | 140 | −4 |
The last observation was reserved as a check rather than used to choose the rate. It is 4 L below prediction. A model that matches earlier data exactly is not guaranteed to predict later values accurately.
A residual is not automatically an instrument error. It can reflect measurement uncertainty, a changing flow, an incorrect initial volume or an inadequate model. The uncertainty of the observations matters when deciding how large a residual is meaningful. Linear Regression develops a formal fitting method; fitting and validation should still be distinguished.
Examine sensitivity before changing the model
Suppose the actual constant inflow could be anywhere from 11.5 to 12.5 L/min, while the initial volume and capacity are treated as fixed. Since fill time is for flow , the fastest and slowest times are
This is a deterministic range based on stated flow bounds, not a statistical confidence interval. It shows which parameter uncertainty affects the answer and by how much.
More detail is not automatically better. A time-varying rate needs evidence or a physical reason, and a fitted polynomial through every observation can behave implausibly between points or outside the measured interval.
Revise one assumption at a time
If the pump rate falls after eight minutes, a piecewise-rate model may be appropriate. If the vessel is not rectangular, volume and height may no longer be proportional. If the initial reading is wrong, an intercept adjustment is more relevant than a new growth law.
Document what changed and why. Recalculate predictions, compare them against independent data where possible, and retain the physical constraints. “Improved fit” and “better explanation” are related but not identical claims.
The same modelling discipline applies to costs, travel and growth. Define the quantities, state the domain, check dimensions, solve, interpret, and test. Do not infer validity solely from a plausible-looking graph.
Practice questions
- A 100 L tank initially contains 10 L and fills at a constant 6 L/min. Find its fill time.
- A model predicts 70 units and an observation is 67. Find the residual under the convention used here.
- Why should the original tank’s uncapped model not be used to report its stored volume after 20 minutes?
Check your answers
- The required addition is 90 L, giving minutes.
- Residual is observed minus predicted: units.
- The unrestricted line gives L, exceeding 160 L capacity. The domain has been exceeded; stopping or overflow must be modelled separately.
Related topics
Use Financial Mathematics for models of dated cash flows, or Growth and Decay for quantities changing by proportional rather than fixed amounts.