At a glance
Pairs · launched with the box problem, due two classes later · one real package, one optimum, one honest accounting
What you are making
The package you brought from home was designed by somebody — and your pair has just been hired to beat them. The brief: a redesign that holds exactly the same volume, fits the client’s shelf (no taller than the shelf gap you measure in class), and uses less material. You finish with a mathematical model — variables named, the volume constraint folded in so material becomes a function of one variable — the optimum located with a derivative and confirmed with a sign argument, a comparison against the real package in square centimetres and in percent, and the honest paragraph: what your “best” ignored, and why the manufacturer might still be right.
Milestones
- The real package measured; its volume and surface area computed, then doubted and measured again
- Variables named and the volume constraint used to write material as a function of one variable, with its sensible domain stated
- Candidate optima located with the derivative and confirmed with a sign argument or Curve Sketching thinking
- Your optimum compared with the real design — material saved stated in and as a percentage
- The honest paragraph: stacking, gripping, branding, the label’s face — what the model never saw
How it is assessed
Per How Marks Work, the reasoning is the product: a modest saving with an airtight argument and an honest accounting outranks a spectacular claim that skipped the domain. On the due date your pair defends the design out loud — including what “best” cost. The Math Journal entry on where your model and the real world part company completes the evidence.
Success criteria
| Quality | What it looks like in your work |
|---|---|
| A constraint obeyed | Volume held exactly; the shelf respected |
| One honest variable | The constraint folded in before deriving |
| A proven optimum | The critical point confirmed, not assumed |
| A fair comparison | Savings stated in units and in percent |
| Costs admitted | What the optimum ignores, said plainly |
If the algebra grows thorns
Fold the constraint in before differentiating, not after — one variable is always tamer than two. And keep the domain in view: a cut, a radius, or a height that goes negative is the model telling you where its world ends.
Curriculum connection
B2.4
solve optimization problems involving polynomial, simple rational, and exponential functions drawn from a variety of applications, including those arising from real-world situations Sample problem: The number of bus riders from the suburbs to downtown per day is represented by , where is the fare in dollars. What fare will maximize the total revenue?
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B2.5
solve problems arising from real-world applications by applying a mathematical model and the concepts and procedures associated with the derivative to determine mathematical results, and interpret and communicate the results Sample problem: A bird is foraging for berries. If it stays too long in any one patch it will be spending valuable foraging time looking for the hidden berries, but when it leaves it will have to spend time finding another patch. A model for the net amount of food energy in joules the bird gets if it spends minutes in a patch is . Suppose the bird takes 2 min on average to find each new patch, and spends negligible energy doing so. How long should the bird spend in a patch to maximize its average rate of energy gain over the time spent flying to a patch and foraging in it? Use and compare numeric, graphical, and algebraic strategies to solve this problem.
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