Optimization

In The Box Problem, every group cut different squares from the corners of the same sheet and folded up a box — and the volumes posted at the boards disagreed wildly. Small cuts made a wide, shallow tray; big cuts made a tall, skinny chimney; somewhere in between, one group’s box beat everyone’s. The question “which cut is best?” is optimization, and it is the payoff of everything since The Derivative: at the very top of the volume curve, the tangent is flat. Maximums hide where the derivative is zero.

From story to function

The calculus in an optimization problem is usually the short part. The real work is translation:

1. Name the variable you control and the quantity you want to maximise or minimise.
2. Write the quantity as a function of the variable — use the constraint to eliminate everything else.
3. State the domain the story allows. A cut of 15 cm from a 24 cm sheet is not a boxy opinion; it is impossible.
4. Differentiate, find critical numbers, and decide which one wins.

For the box: $V(x)=x(24-2x{)}^2$ on $0<x<12$, and $V^{\prime }(x)=(24-2x)(24-6x)$ is zero at $x=4$ — the cut the winning group found by folding.

Finding and auditing the peak

A critical number is a candidate, not a verdict. The audit is where the thinking lives:

A flat tangent is not always a summit

$V^{\prime }(x)=0$ also happens at valley bottoms, and the extreme value you want sometimes sits at an endpoint of the domain instead. Confirm every candidate: check the sign of $V^{\prime }$ on each side (uphill then downhill means maximum), or evaluate the function at every candidate and endpoint and compare. Then reread the question — an answer of $x=4$ is incomplete if the question asked for the volume.

A wrong candidate that survives to your final answer is not a small slip; it is the whole problem. This is a place where checking is the mathematics — Mistakes Are Data is about exactly this habit.

Optimization is also where models earn their keep: bus fares that maximise revenue, packaging that minimises material, a foraging bird budgeting its minutes. The curriculum calls this applying a mathematical model, and it is the shape of the The Packaging Brief task — a real package, real constraints, and a defended recommendation. A Would You Rather instinct helps before any algebra starts: roughly where should the best answer live, and why? Estimate first, optimize second, and let the estimate audit the calculus. Optimization Practice has the full range, from warm-ups to the bird.

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 $1200(1.15{)}^{-x}$, where $x$ 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 $t$ minutes in a patch is $E=\frac{3000t}{t+4}$. 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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