FAQ
If more than one input value gives the same maximum or minimum, state each permissible value clearly and explain that the optimised quantity is identical for all of them.
Then specify which of the values are feasible within the problem’s context. For example, negative dimensions, unrealistic times, or values outside the domain must be excluded even if they appear mathematically valid.
Your units should match the level of precision implied by the problem. If measurements are given to the nearest centimetre, report your answer using centimetres, not rounded metres.
Where appropriate, include derived units clearly. For example, if a length is optimised first and then used to compute an area, the area must be reported in square units.
Students often state only the optimised number without explaining what it represents.
Another frequent error is mixing up the independent variable and the quantity being optimised, such as giving the dimension that produces the maximum instead of the maximum itself—or vice versa.
Use wording that links the mathematical test to the real context.
For example:
• State the behaviour of the function (increasing or decreasing) around the critical value.
• Briefly mention why this behaviour means the outcome is best or least within the real constraints of the scenario.
It is good practice to refer back to the constraint if it helps clarify the meaning of the answer.
For example, you might mention that a dimension is optimal “under the requirement that the perimeter is fixed at 40 metres”. This helps the reader understand the conditions under which the result applies.
Practice Questions
Question 2 (4–6 marks)
A farmer wants to enclose the greatest possible area using 40 metres of fencing to form a rectangular field. It is found using calculus that the area is maximised when the rectangle is a square.
(a) State the dimensions of the rectangle that maximise the area, including correct units.
(b) Clearly state what quantity has been maximised.
(c) Explain why these dimensions give a maximum and not a minimum, referring to the context of the problem.
(d) The farmer suggests using only 20 metres of fencing instead. Explain how this change in constraint affects the interpretation of any optimised solution.
Question 2
(a) (1–2 marks)
States that a square with side length 10 metres maximises the area.
• Correct side length (10) (1 mark)
• Correct unit (metres) and clarity that both sides are 10 m (1 mark)
(b) (1 mark)
States clearly that the quantity maximised is the enclosed area of the field.
(c) (1–2 marks)
Explains that the calculus result corresponds to a maximum because either:
• the derivative changes from positive to negative (1 mark)
• or because among feasible dimensions under the fixed perimeter, the square gives the largest possible area, relating this to the context (1 mark)
(d) (1 mark)
Explains that reducing the available fencing changes the constraints, so the optimal dimensions must be recalculated. The new maximum area will be smaller because the total boundary is reduced.
