Optimization Problems

1. Draw a picture and identify variables.
2. Write the objective function (what to maximize/minimize).
3. Write a constraint equation to reduce to one variable.
4. Differentiate and set $= 0$.
5. Verify it's a max or min (first/second derivative test).
6. Answer the question asked.

Maximum Area (Fixed Perimeter)

Rectangle with perimeter 40: $2l + 2w = 40$ → $w = 20 - l$. $A = l(20-l)$. $A' = 20 - 2l = 0$ → $l = 10$. Square: $10 \times 10$.

Minimum Material (Fixed Volume)

Open box volume $V = x^2h$. Surface area $S = x^2 + 4xh$. Substitute $h = V/x^2$ and minimize.

Fence 3 sides of a rectangle (river on 4th side). 200 m of fence. Maximize area.

$2x + y = 200$ → $y = 200 - 2x$. $A = x(200-2x) = 200x - 2x^2$.

$A' = 200 - 4x = 0$ → $x = 50$. $y = 100$. Max area = 5000 m².