Maths › Algebra and functions › Functions in modelling
Functions in modelling
By now the course has a shelf of function families, and modelling is a matter of choosing the right one, fitting its constants to the situation, and stating where it stops working. Tides call for trig and cooling calls for exponentials, and the choice itself is what gets examined.
Builds on Log graphs and exponential models and Trigonometric graphs and equations.
IN THIS TOPIC
- Match a situation's behaviour to the function family that describes it.
- Fit and interpret a model's constants in context, and state clearly where the model fails.
COMMON MISCONCEPTION
The best model is the one that fits the data most closely.
Choosing the family
Each function family in the course has a behavioural signature, and diagnosis comes before fitting. Behaviour that repeats at a fixed interval points to trig. A rate proportional to the current amount points to an exponential. Two quantities whose product stays constant point to a reciprocal. Families combine too, a constant plus a trig term for tides, a constant plus a decaying exponential for cooling.
| Behaviour observed | Family to reach for |
|---|---|
| repeats every fixed interval | trigonometric: a + b sin(kt) |
| rate proportional to amount | exponential: Ae to the kt |
| product of the two quantities constant | reciprocal: k/x |
| steady rate of change | linear: mx + c |
WORKED EXAMPLE
Reading a tide model
The depth of water in a harbour is modelled by h = 5 + 2.4 sin (30t)°, with t in hours after midnight. Find the period of the tide, the greatest and least depths, and the first time of high tide.
The sine completes a cycle when 30t reaches 360, so the period is 12 hours.
Sine runs between ±1, so h runs from 5 − 2.4 = 2.6 m to 5 + 2.4 = 7.4 m.
High tide needs sin (30t)° = 1, first at 30t = 90, so t = 3, three in the morning.
The 5 is the mean depth, the 2.4 is the tidal amplitude, and the 30 sets the clock. Every constant answers to a physical question, and that is what separates a model from a formula.
Fitting, and knowing the limits
A fitted model earns trust only inside the conditions it was built for; closeness of fit alone earns none. A curve through every data point can still belong to the wrong family, and the wrong family extrapolates into nonsense. What gets examined is naming the assumption a model makes, spotting where reality breaks it, and proposing the refinement.
GUIDED PRACTICE
A cooling drink, read in full
A drink's temperature is modelled by T = 20 + 60e−0.05t, t in minutes. State the initial temperature, explain the physical meaning of the 20, and find when T reaches 50°, before opening the working.
Show the working
At t = 0 the exponential is 1, so T starts at 20 + 60 = 80°.
As t grows the exponential dies away and T flattens onto 20°, the room's temperature. The drink cools towards its surroundings and never below them.
Setting T = 50 gives e−0.05t = ½, so t = ln 2/0.05 = 13.9 minutes.
A bare decay model T = 80e−kt would have the drink approaching 0°, which no room allows. The added constant encodes a physical fact the simple model missed, which is what a refinement is for.
INDEPENDENT PRACTICE
Criticise and refine
Hours of daylight in a town are modelled by D = 12 + 4.2 sin (30m)°, with m in months after the spring equinox. State what the model predicts for the longest day, and give one limitation and one refinement.
Show the working
The longest day is 12 + 4.2 = 16.2 hours, three months in, at midsummer.
One limitation. Real daylight is not perfectly sinusoidal, and months are of unequal length, so the model drifts against the calendar across the year.
One refinement. Fit the period in days instead of months, or set the amplitude from the town's latitude. Naming a specific, checkable improvement is what the mark scheme wants, and “collect more data” on its own is not one.
ASSESSMENT FOCUS
- Diagnose the family from the behaviour before touching any constants. One sentence naming the signature earns the first mark.
- Interpret every constant with units and a physical meaning. “Initial” means t = 0 and long-term behaviour means the asymptote, and both are one-line evaluations.
- A limitation names the assumption that fails and says where. A refinement changes the model. “Collect more data” does neither.
CHECK YOURSELF
The value of a machine is modelled by V = 500 + 7500e−0.25t pounds after t years. State the initial value, the long-term value, and what the 500 represents.
Show a hint
Evaluate at t = 0, then let the exponential die.
Show the answer
At t = 0, V = 500 + 7500 = £8000.
As t grows, the exponential vanishes and V flattens onto £500.
The 500 is the machine's scrap or residual value, the lower limit the model settles towards as t grows, and it is the refinement that separates this from bare exponential decay.
Behaviour picks the family. Repetition is trig, a rate proportional to the amount is exponential, a constant product is reciprocal.
Fit the constants, read them in units, and say where the model breaks.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
Or read them with their worked answers on the functions in modelling questions page.
CHECK YOUR PROGRESS
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- Match a situation's behaviour to the function family that describes it.
- Fit and interpret a model's constants in context, and state clearly where the model fails.
Open the full revision checklist to see every objective in the course in one place.