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Optimization

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Optimization - Complete Interactive Lesson

Part 1: Setting Up Optimization Problems

Optimization

Part 1 of 7 — Setting Up Optimization Problems

Topic Overview

Part	Topic
1	Setting up optimization problems
2	Geometric optimization
3	Applied & business optimization
4	3D optimization (cylinders & cones)
5	Distance & mixed optimization
6	AP-style workshop
7	Comprehensive assessment

The 5-Step Optimization Strategy

$\boxed{\text{1. Variables} \to \text{2. Objective} \to \text{3. Constraint} \to \text{4. Critical pts} \to \text{5. Verify}}$

Step	Action	Example
1. Define variables	Label unknowns, draw a diagram	$x =$ width, $y =$ length
2. Write objective	Function to max/min	$A = xy$
3. Apply constraint	Eliminate one variable

Worked Example: Rancher Fencing

A rancher has 200 m of fencing to enclose a rectangle along a river (no fence on the river side). Find the maximum area.

Let $x =$ width, $y =$ length along river.

Constraint: $2x + y = 200 \Rightarrow y = 200 - 2x$

Objective: $A(x) = x(200-2x) = 200x - 2x^2$

$A'(x) = 200 - 4x = 0 \quad \Rightarrow \quad x = 50$

$A''(x) = -4 < 0$ — concave down — maximum

$y = 200-100 = 100$ . $\boxed{A_{\max} = 50 \times 100 = 5000 \text{ m}^2}$

Key Fact: The optimal rectangle along a wall always has the side parallel to the wall equal to twice the perpendicular side.

Practice — Setting Up 🎯

Identify the setup. 🔍

Solve. ✍️

Key Takeaways — Part 1

Define variables clearly and draw a diagram
The objective is what you maximize/minimize
The constraint reduces to one variable
Use the Second Derivative Test or endpoint analysis to verify
AP Tip: Always state the domain of the objective function

Part 2: Geometric Optimization

Optimization

Part 2 of 7 — Geometric Optimization

Classic Geometry Problems

Problem Type	Setup
Box from sheet	Cut squares of side $x$ from corners, fold up
Rectangle in parabola	Vertices at $(\pm x, 0)$ and $(\pm x, f(x))$

Part 3: Cost & Revenue Optimization

Optimization

Part 3 of 7 — Applied & Business Optimization

Business Terminology

Term	Formula	Meaning
Revenue	$R(x) = x \cdot p(x)$	Income from selling $x$ units at price

Part 4: 3D Optimization (Cylinders & Cones)

Optimization

Part 4 of 7 — 3D Optimization (Cylinders & Cones)

3D Shape Formulas

Shape	Volume	Surface Area
Cylinder (closed)	$\pi r^2 h$	$2\pi r^2 + 2\pi rh$

Part 5: Distance & Angle Optimization

Optimization

Part 5 of 7 — Distance & Mixed Problems

Distance Optimization

When minimizing the distance from a point $(a,b)$ to a curve $y = f(x)$ :

$D^{2} = (x - a)^{2} + ($

Part 6: AP-Style Workshop

Optimization

Part 6 of 7 — AP-Style Workshop

AP FRQ Optimization Patterns

Pattern	What They Ask	Key Steps
Geometric	"Find dimensions that maximize/minimize..."	Draw diagram, label, constrain
Applied	"At what rate/quantity is profit maximized?"	$P = R - C$ , set $P' = 0$

Part 7: Comprehensive Assessment

Optimization

Part 7 of 7 — Comprehensive Assessment

Optimization Strategy Reference

Step	Action	Check
1	Draw and label diagram	Variables defined?
2	Write objective function	What are you optimizing?
3	Apply constraint	Down to one variable?
4	Find critical points	$f'(x) = 0$ solved?

Worked Example: Open-Top Box

An open-top box is made by cutting squares of side $x$ from the corners of a $12 \times 8$ sheet and folding up.

$V(x) = x(12-2x)(8-2x)$

Domain: $0 < x < 4$

$V = 4x^3 - 40x^2 + 96x$ $V'(x) = 12x^2 - 80x + 96 = 4(3x^2 - 20x + 24)$

$x = \frac{20 \pm \sqrt{400-288}}{6} = \frac{20 \pm \sqrt{112}}{6}$

$x \approx 1.57$ (only solution in domain). $\boxed{V_{\max} \approx 67.6 \text{ cubic units}}$

AP Tip: When asked to "set up but do not solve," write $V(x)$ , state the domain, and show $V'(x) = 0$ . You earn full credit without solving the quadratic.

Worked Example: Closest Point

Find the point on $y = \sqrt{x}$ closest to $(3, 0)$ .

Minimize $D^2 = (x-3)^2 + (\sqrt{x})^2 = (x-3)^2 + x$

$\frac{d}{dx}D^2 = 2(x-3) + 1 = 2x - 5 = 0 \quad \Rightarrow \quad x = \frac{5}{2}$

Point: $\left(\frac{5}{2},\, \sqrt{\frac{5}{2}}\right)$

Key Fact: Always minimize $D^2$ instead of $D$ — it avoids square roots and gives the same critical points.

Practice — Geometric Optimization 🎯

Identify the correct approach. 🔍

Calculate. ✍️

Key Takeaways — Part 2

Box-cutting problems: $V = x(L-2x)(W-2x)$ , domain $0 < x < \min(L,W)/2$
Minimize $D^2$ for closest-point problems
Inscribed rectangle under a curve: $A = 2x \cdot f(x)$
Always check domain endpoints for absolute max/min

$\boxed{\text{Max profit where } R'(x) = C'(x)}$

Worked Example: Widget Company

A company sells widgets at price $p = 100 - 2x$ per widget. Cost: $C(x) = 200 + 5x$ . Find maximum profit.

$R(x) = x(100-2x) = 100x - 2x^2$ $P(x) = R(x) - C(x) = 100x - 2x^2 - 200 - 5x = -2x^2 + 95x - 200$ $P'(x) = -4x + 95 = 0 \quad \Rightarrow \quad x = 23.75$

Since $x$ must be whole: $P(24) = -2(576) + 95(24) - 200 = -1152 + 2280 - 200 = 928$ .

$\boxed{P_{\max} = \$ 928 \text{ at } x = 24 \text{ widgets}}$

AP Tip: On the AP exam, optimization word problems may use business language. Know the formulas for $R$ , $C$ , $P$ , and marginal quantities.

Practice — Applied Optimization 🎯

Classify the quantity. 🔍

Key Takeaways — Part 3

Profit $= R - C$ ; max profit when $R'(x) = C'(x)$
Average cost $\bar{C} = C(x)/x$ ; minimize by setting $\bar{C}' = 0$
Revenue $R(x) = x \cdot p(x)$ where $p(x)$ is the demand function
When units must be integers, check both nearest whole numbers

Worked Example: Minimize Surface Area

A closed cylinder has volume $V = 1000$ cm $^3$ . Find the radius that minimizes surface area.

Constraint: $\pi r^2 h = 1000 \Rightarrow h = \frac{1000}{\pi r^2}$

Objective: $S = 2\pi r^2 + 2\pi rh = 2\pi r^2 + \frac{2000}{r}$

$S'(r) = 4\pi r - \frac{2000}{r^2} = 0$ $4\pi r^3 = 2000 \quad \Rightarrow \quad r = \left(\frac{500}{\pi}\right)^{1/3} \approx 5.42 \text{ cm}$

Then $h = \frac{1000}{\pi(5.42)^2} \approx 10.84 \approx 2r$ .

$\boxed{\text{Optimal closed cylinder: } h = 2r}$

Key Fact: The optimal closed cylinder always has $h = 2r$ (height equals diameter). For an open-top cylinder, the optimal ratio is $h = r$ .

Practice — 3D Optimization 🎯

Identify the setup. 🔍

Calculate. ✍️

Key Takeaways — Part 4

3D optimization: same 5-step process with volume/surface area formulas
Closed cylinder: optimal when $h = 2r$
Open-top cylinder: optimal when $h = r$
Always express $S$ or $V$ in one variable using the constraint

D^{2} = (x - a)^{2} + (f (x) - b)^{2}

Key Fact: Always minimize $D^2$ — it shares the same critical points as $D$ and avoids square roots.

Worked Example: Wire-Cutting

A wire of length 20 is cut into two pieces. One is bent into a square, the other into a circle. What cut minimizes total area?

Let $x =$ circumference of circle, $20-x =$ perimeter of square.

Circle: $r = \frac{x}{2\pi}$ , area $= \frac{x^2}{4\pi}$

Square: side $= \frac{20-x}{4}$ , area $= \frac{(20-x)^2}{16}$

$A(x) = \frac{x^2}{4\pi} + \frac{(20-x)^2}{16}$

$A'(x) = \frac{x}{2\pi} - \frac{20-x}{8} = 0$

$\frac{4x}{8\pi} = \frac{\pi(20-x)}{8\pi} \quad \Rightarrow \quad 4x = 20\pi - \pi x$

$x = \frac{20\pi}{\pi + 4} \approx 8.80$

Special Optimization Patterns

Pattern	Key Idea
Max/min with absolute value	Split into cases
Optimization on closed interval	Check critical points AND endpoints
Profit with discrete units	Check both integers near the critical point
Constrained by two inequalities	Domain may be restricted

Practice — Mixed Problems 🎯

Choose correctly. 🔍

Key Takeaways — Part 5

Minimize $D^2$ for closest-point problems (avoids square roots)
Wire-cutting/splitting problems: express total in one variable
Closed interval: always check endpoints AND critical points
Discrete constraints: check both nearby integers

Full Worked AP Problem

A rectangle has one side on the $x$ -axis, the upper two vertices on $y = 4 - x^2$ .

(a) Express the area $A$ in terms of $x$ .

(b) Find the value of $x$ that maximizes $A$ .

(c) Justify that your answer gives a maximum.

Solution (a): Vertices at $(\pm x, 0)$ and $(\pm x, 4-x^2)$ . Width $= 2x$ , height $= 4-x^2$ . $A(x) = 2x(4-x^2) = 8x - 2x^3, \quad 0 < x < 2$

Solution (b): $A'(x) = 8 - 6x^2 = 0 \Rightarrow x^2 = 4/3 \Rightarrow x = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}$

Solution (c): $A''(x) = -12x$ . At $x = 2\sqrt{3}/3$ : $A'' = -12 \cdot \frac{2\sqrt{3}}{3} < 0$ .

Since $A'' < 0$ at the critical point, $A$ is concave down there, so this gives a maximum.

$A_{\max} = 2 \cdot \frac{2\sqrt{3}}{3}\left(4 - \frac{4}{3}\right) = \frac{4\sqrt{3}}{3} \cdot \frac{8}{3} = \frac{32\sqrt{3}}{9}$

AP Tip: On justification, you must state the Second Derivative Test result AND explicitly conclude "maximum" or "minimum." Just computing $f''$ is not enough.

AP-Style Practice 🎯

AP Problem. ✍️

Key Takeaways — Part 6

AP FRQs require complete justification for max/min
"Write but do not solve" earns credit for setup
Second Derivative Test is the standard justification
On closed intervals, compare all candidates (critical points + endpoints)

Mistake	Fix
Not stating the domain	Write $0 < x < \text{bound}$
Forgetting to verify max vs min	Always use 2nd derivative test
Wrong constraint	Re-read what's "fixed" or "given"
Ignoring endpoints on $[a,b]$	Compare $f$ at all candidates
Not labeling units	Include m, cm $^2$ , etc.

Quiz Set 1 — Core Skills 🎯

Quiz Set 2 — Advanced Problems 🎯

Final review. 🔍

Final Challenge. ✍️

🎉 Topic Complete!

You've mastered Optimization:

Part	Topic	Status
1	Setting up optimization problems	✅
2	Geometric optimization	✅
3	Applied & business optimization	✅
4	3D optimization (cylinders & cones)	✅
5	Distance & mixed problems	✅
6	AP-style workshop	✅
7	Comprehensive assessment	✅

Key Fact: The 5-step optimization strategy (variables → objective → constraint → critical points → verify) works for every optimization problem. On the AP exam, always justify your answer using the Second Derivative Test or endpoint comparison.

Optimization

Optimization - Complete Interactive Lesson

Part 1: Setting Up Optimization Problems

Optimization

Topic Overview

The 5-Step Optimization Strategy

Worked Example: Rancher Fencing

Key Takeaways — Part 1

Part 2: Geometric Optimization

Optimization

Classic Geometry Problems

Part 3: Cost & Revenue Optimization

Optimization

Business Terminology

Part 4: 3D Optimization (Cylinders & Cones)

Optimization

3D Shape Formulas

Part 5: Distance & Angle Optimization

Optimization

Distance Optimization

Part 6: AP-Style Workshop

Optimization

AP FRQ Optimization Patterns

Part 7: Comprehensive Assessment

Optimization

Optimization Strategy Reference

Worked Example: Open-Top Box

Worked Example: Closest Point

Key Takeaways — Part 2

Worked Example: Widget Company

Key Takeaways — Part 3

Worked Example: Minimize Surface Area

Key Takeaways — Part 4

Worked Example: Wire-Cutting

Special Optimization Patterns

Key Takeaways — Part 5

Full Worked AP Problem

Key Takeaways — Part 6

Common AP Mistakes

🎉 Topic Complete!