The idea, step by step
Describe assumptions that affect the answer
A linear model assumes constant per-unit rates and additive resource use over the chosen range. It may assume that all output can be sold, resources are simultaneously available and decisions have no setup interactions. Tie each assumption to the actual situation rather than saying only that the model is realistic.
An optimum is conditional on the model
Check demand, minimum orders, staffing, setup times and uncertain data. Change justified inputs and re-optimise when needed. An optimum using all of a resource can become infeasible after a small loss of capacity. Report suitable precision and what additional real information would make the decision more reliable; correct algebra alone does not validate a model.
A common trap
Correct optimisation proves the best real-world decision.
Instead: It proves a best decision within the stated model and data. The real-world assumptions still need checking.
See the thinking
A missing sales limit
A model recommends 40 deluxe items, but reliable orders are at most 25. Explain the needed correction.
- If y counts deluxe items, include .
- The previous best point may now be infeasible.
- Solve the revised model instead of simply asserting that 40 remains optimal.
A constant-rate assumption
A bus-hire model is . Why might it fail for an overnight trip?
- The fixed fee and per-kilometre rate describe the original operating conditions.
- An overnight trip might require extra driver wages or accommodation.
- Use supplied additional information to revise the rule or construct a piecewise model.
Check your understanding
A model allows 2.6 coaches. What condition is likely missing?
Show answer
A whole-number restriction on coach counts.
Hiring decisions use indivisible vehicles.
Your turn
6 questions · 12 marks · Work at your own pace.
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 ↗