1. The "Primary" Function: What are you trying to Maximize or Minimize? Write its formula first.
2. The "Constraint": What is the limit? (e.g., total length, total cost). Write its equation.
3. The Substitution: Use the Constraint to turn your Primary into a Single Variable.
4. The Peak Check: Find the derivative, set to zero, and solve.
A shepherd has 60 meters of fencing. He wants to create a rectangular pen against a stone wall (so he only needs to fence 3 sides).
Build the Model: Let $x$ be the width. Express the Area ($A$) as a function of $x$.
Find the Peak: Find the width ($x$) that gives the maximum area.
Calculate the Harvest: What are the dimensions of the pen, and what is the maximum area?
If the shepherd used all 4 sides of the fence (not against a wall), the optimal shape would be a Square (15x15 = 225m²). But against a wall, the optimal shape is not a square (it's 15x30 = 450m²). Why does having a "God-given boundary" (the wall) allow us to have so much more room with the same amount of effort?
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Find two positive numbers whose sum is 20 and whose product is a maximum.
Find two positive numbers whose product is 100 and whose sum is a minimum.
A ministry needs to design a box with a square base and no top. The box must have a volume of 32 cubic feet.
Volume: $x^2 \cdot h = 32 \implies h = 32/x^2$.
Surface Area (Material): $S = x^2 + 4xh$.
Task: Find the dimensions of the box ($x$ and $h$) that will use the Minimum amount of material.
Objective: Explain Optimization to a younger student using a loop of string.
The Activity:
1. Make a 12-inch loop of string.
2. Have them make different rectangles with it on a table.
3. Ask: "Which one looks like it holds the most toys?"
4. Prove it by counting blocks inside.
The Lesson: "Rules (the string) aren't meant to stop us; they are meant to help us find the 'Best' way to use what we have."
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