Understand the idea.
Describe assumptions that affect the answer
A linear model assumes constant per-unit rates and additive totals over a stated range. In production, this may mean each item always uses the same labour and every item produced can be sold. These assumptions directly affect whether a recommended mix is achievable.
Name the assumption and its consequence in the actual situation. For example, limited demand requires a sales bound; a setup time may require additional constraints. Saying only that a model is realistic does not identify what must be checked.
An optimum is conditional on the model
An optimal solution is conditional on the model and its data. Correct algebra establishes the best permitted decision within that model; it does not establish that demand, prices or resource estimates will hold in practice.
When reliable new information changes a constraint or cost, revise the model and solve it again. Check sensitivity near fully used resources: even a small reduction in capacity can make the old recommendation infeasible. Report uncertainty and sensible precision.
See the method in action.
A missing sales limit
A model recommends 40 deluxe items, but reliable orders are at most 25. Explain the needed correction.
Worked solution
- 1
Define the quantity affected by demand.
The sales limit must be applied to the variable representing deluxe items.
- 2
Add the supported demand bound.
Reliable orders support at most 25 sales, so the feasible model must enforce that limit.
- 3
Test the previous recommendation.
The old recommendation violates the new restriction and is no longer feasible.
- 4
Re-optimise with all constraints retained.
Compare decisions in the revised feasible region. The new best mix cannot be calculated without the remaining constraints and objective.
Add the supported demand bound and re-optimise.
A constant-rate assumption
A bus-hire model is . Why might it fail for an overnight trip?
Worked solution
- 1
Interpret the original cost rule.
It assumes a fixed AUD 150 charge and a constant AUD 3 per kilometre under the original hire conditions.
- 2
Identify a condition changed by an overnight trip.
Additional driver time or accommodation may create a cost not represented by distance alone.
- 3
Identify the information needed for a revision.
Obtain the actual overnight charges and applicable conditions before revising the fixed fee, adding a term or using a piecewise rule.
- 4
State the limitation of the existing prediction.
The original rule may not apply outside its operating range. No revised numerical cost follows without the missing charges.
The original costs may not apply outside the model's original range.
Try it yourself.
Check your understanding
A model allows 2.6 coaches. What condition is likely missing?
Show answer & reasoning
A whole-number restriction on coach counts.
Coaches are hired as whole vehicles. The model needs a non-negative integer condition on the coach count; whether 2 or 3 works must then be checked against the capacity and cost constraints.
Keep practising
6 worksheet questions · 12 marks
Read the questions online
- A model assumes every cake made is sold. Suggest one useful extra restriction.2 marks
- An optimum uses all 50 staff hours. An absence removes 4 hours. What must change?2 marks
- A plotted optimum appears near . Why solve exact boundary equations before choosing integer counts?2 marks
- A higher-price product also has higher unit cost. Is it necessarily better for profit?2 marks
- A model predicts unlimited profit with no capacity or demand bound. What does that suggest?2 marks
- A small change in one profit coefficient changes the best plan. What should a report explain?2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- A model assumes every cake made is sold. Suggest one useful extra restriction.2 marks
Use evidence of demand to limit production.
A maximum demand or sales bound. - An optimum uses all 50 staff hours. An absence removes 4 hours. What must change?2 marks
The former optimum may be infeasible.
Use a 46-hour bound and re-optimise. - A plotted optimum appears near . Why solve exact boundary equations before choosing integer counts?2 marks
Find the exact intersection, then test integer choices against the original inequalities.
Graph rounding can distort the true corner and feasibility. - A higher-price product also has higher unit cost. Is it necessarily better for profit?2 marks
Compare unit contribution and use of scarce resources, not selling price alone.
No. - A model predicts unlimited profit with no capacity or demand bound. What does that suggest?2 marks
Add justified practical information rather than an arbitrary cap.
The model is unbounded and may omit a real restriction. - A small change in one profit coefficient changes the best plan. What should a report explain?2 marks
State the changed data and result; this is not necessarily an arithmetic error.
The recommendation is sensitive to that input.
Curriculum reference & lesson scope
SACE Board of South Australia
Stage 2 General Mathematics: Modelling with Linear Relationships, a compulsory school-assessed topic. The external examination scope differs from the full course.
Official curriculum source ↗