Lesson 7: Optimisation Problems & Real-World Modelling
The hard part of an optimisation question is not the calculus. It is turning a sentence into one function of one variable — the constraint step is where the Excellence marks are won and lost.
🎯 Ngā Whāinga Akoranga | Learning Intentions
Constructing mathematical models from real-world constraints to find maximum or minimum values (optimising volume, surface area, cost, distance, profit).
Formulate single-variable objective functions from word problems, differentiate to find optimal stationary points, and prove global max/min using boundary checks.
🎥 Media Anchor & Pedagogical Scaffold
Optimisation: Cost of Materials — Khan Academy
Video (12 min 39 sec, Khan Academy): A full worked optimisation problem: minimising the cost of the materials for a container. It is the whole method end to end — name the quantity being optimised, write the constraint, reduce to one variable, differentiate, solve, and check. This is exactly the shape of an NCEA Excellence optimisation question.
🧠 1. Before Viewing (Activate & Predict)
How do industrial engineers use calculus to design packaging that minimises material cost while maximising container volume?
👁️ 2. During Viewing (Watch With a Job)
Watch the full 12 min 39 sec clip. In your calculus logbook, record:
- The setup: Write down what quantity is being minimised and what the constraint is, in the video's own terms, before any calculus happens. Most lost marks in this topic are lost here.
- The reduction step: Copy the line where two variables become one. Mark which equation was substituted into which — that substitution is the whole trick.
- The verification: Note how the video confirms the stationary point really is a minimum rather than assuming it.
🗣️ 3. After Viewing & Kaiako Move (Process & Apply)
Kaiako Move: Add the boundary check the clip does not need: on a restricted domain [a, b], the absolute maximum or minimum may sit at an endpoint, so evaluate f at a and b as well as at every stationary point. Then work the two classic NCEA Excellence problems — minimising the material in a cylindrical tin, and maximising a fenced rectangular paddock.
Immediate Task: Add to your Level 3 Calculus revision logbook (section 7): Optimisation Word Problem & Engineering Model Set.
⚡ Whakaoho | Do Now: From Words to One Variable (5 mins)
A kura is building a rectangular maara kai bed and has 20 m of timber edging for its four sides. Write the growing area in terms of one variable only. Do not maximise it yet. The hard part of optimisation is this step, not the differentiation.
📖 Activity 1: Optimising, Including at the Boundary (20 mins)
Finish your Do Now bed: maximise the growing area and justify the maximum with the second derivative. Then the case that separates Merit from Excellence: find the absolute maximum of f(x) = −x² + 4x on the interval [0, 3]. Solve f′(x) = −2x + 4 = 0 to get x = 2, then evaluate f(0) = 0, f(2) = 4 and f(3) = 3. Now change the interval to [0, 1] and repeat. The maximum moves to an endpoint, and no stationary point will tell you that. On a closed interval you must always check the endpoints.
📝 Activity 2: Exam-Style Practice — Model, Optimise, Interpret (15 mins)
An open-topped box is made from a 20 cm by 20 cm sheet by cutting squares of side x from each corner. Find the x that maximises volume. For Merit, state the domain of x and justify the maximum. For Excellence, explain why x = 10 is excluded and what the volume does as x approaches that value.
For Excellence — the same shape of problem, a different context. A waka ama (outrigger canoe) crew wants to find the cruising speed that costs the least energy over a fixed race distance. This is a scenario built to be worked by hand — the power figures are illustrative, not measured hull or paddler data. Water drag scales with the square of speed, so the power needed to overcome it scales with the cube of speed; in this scenario the crew's total power output at speed v (m/s) is modelled as P(v) = v³ + 54 watts, where the constant 54 W stands in for the steady cost of maintaining stroke rhythm and balance, which does not depend on speed. Energy used per metre travelled is then e(v) = P(v) ÷ v = v² + 54/v.
- Find e′(v) and the stationary point of e. (You should get v = 3 m/s, with e″(v) > 0 there, confirming a minimum.)
- At v = 3, e(3) = 27 (scenario units). Show that paddling faster, at v = 4, costs more energy per metre — e(4) = 29.5 — even though the crew finishes the race sooner, and explain in one sentence why that trade-off exists.
This is optimisation exactly as in the box problem — one function of one variable, one stationary point, one second-derivative check. Only the units change, from square centimetres to watt-seconds per metre.
🎫 Exit Ticket: Set It Up, Do Not Solve It (5 mins)
A second maara kai bed runs along an existing shelter belt, so only three sides need edging. There are 120 m of edging. (a) Write the growing area as a function of one variable. (b) Do not solve — state what you would differentiate, and why.
🏫 Kaiako Planning & Pedagogy Notes
NCEA Level 3 alignment: Direct preparation for NCEA Level 3 Achievement Standard AS 91578, Apply differentiation methods in solving problems (external, 6 credits), against The New Zealand Curriculum (2007) Mathematics and Statistics Level 8 — Calculus. Emphasise complete algebraic working and a conclusion written in the context of the question, which is what separates Merit from Achieved.
Materials: Graphics calculator, Desmos, scrap paper for diagrams (insist on a labelled diagram before any algebra), calculus logbook, and the handout Calculus Applications in Environmental Modelling — its kiwi-population and mahinga-kai wetland problems are the applied version of this lesson.
Pacing (63 mins): Do Now 5 · media anchor 18 (the clip plus the before- and after-viewing prompts) · Activity 1 20 · Activity 2 15 · exit ticket 5. The clip is 12 min 39 sec, the longest before the capstone. In a 50-minute period, set it as pre-viewing homework and open with the Do Now instead; the lesson then runs to about 45 minutes.
Formative assessment — what to look for: The exit ticket deliberately stops before the calculus. If ākonga can produce A(x) = x(120 − 2x) and say they would differentiate A with respect to x and set it to zero, the setup is secure and everything after it is Lesson 4 revision. If they cannot, more differentiation practice will not help — reteach the constraint step.
Differentiation. Entry: Give the constraint equation; ākonga do only the substitution down to one variable. On level: Full problems from the word description, with the second derivative test to confirm. Extension: A closed cylindrical tin must hold 500 cm³. Minimise the surface area, then show that the answer h = 2r does not depend on the 500 at all, and explain why.