Geometric Methods for Design Problems

Deck 1 of 2: Modeling Cycle and Optimization

Learning Objectives for This Lesson

By the end of this lesson, you will be able to:

1. Translate requirements into geometric equations
2. Apply the five-step modeling cycle
3. Solve material optimization for cylinders and boxes
4. Check whether a mathematical solution is practical
5. Explain why real designs deviate from optimal models

You Are Starting a Candle Company

Design a candle with three requirements:

• Hold exactly 250 mL of wax (250 cm³)
• Fit in a 12 cm cube gift box
• Minimize wax usage (cost)

Question: What shape? What dimensions? How do you decide?

The Five Steps of the Modeling Cycle

1. Define — What are we designing? What must it do?
2. Constrain — What limits exist? (size, cost, weight)
3. Set Up — Write constraints and objective as equations
4. Explore — Try values; compute objective for each
5. Interpret — Does the solution make sense practically?

Candle Problem: Steps 1 and 2

Step 1 — Define: Cylindrical candle, cm³

Step 2 — Constrain:

• Fits in 12 cm cube: cm and cm
• Objective: minimize surface area (wax cost)

Constraints define what is possible. The objective picks the best.

Candle Problem: Step Three — Set Up

Volume links and — two variables become one:

Once is chosen, is determined.
SA becomes a function of alone.

Candle Problem: Step Four — Explore

SA = 2πr² + 2πrh — substitute constraint for :

Minimum SA near cm within constraints.

Candle Problem: Step Five — Interpret

Winner: cm, cm

Verify:

• Diameter 6 cm < 12 cm ✓
• Height 8.84 cm < 12 cm ✓
• Volume cm³ ✓

Math gives a solution. Interpret checks it's a good one.

Check-In: Setting Up a Design Problem

A storage shed must:

• Hold at least 10 m³
• Fit through a gate that is 3 m wide
• Use minimum material (minimize SA)

Identify:
a) Two constraints (as inequalities)
b) The objective (as a phrase or formula)

Think before advancing.

Answer: Constraints and Objective for Shed

Constraints (Step 2):

Objective (Step 3):

Minimize surface area: SA = 2ℓw + 2ℓh + 2wh

Many valid constraint interpretations exist —
depth, height limits depend on what the design specifies.

From Any Design to the Best Design

The modeling cycle finds valid designs.

Optimization finds the best design within constraints.

Key question: among all cylinders with volume 400 cm³,
which dimensions give the minimum surface area?

→ This is the classic can optimization problem

Volume Constraint Reduces Two Variables to One

For a cylinder with fixed volume V = 400 cm³:

• Choosing determines entirely
• Two-variable problem collapses to one variable

This is why optimization is possible:
once you pick , everything else follows.

Setting Up the Can Optimization Problem

Volume cm³ → constraint links and :

Building the Can Optimization Table

Minimum surface area near cm, cm

The SA vs r Curve Shows the Minimum

Minimum SA occurs at cm — the bottom of the U-curve.

Optimal Cylinder Has Height Equal to Diameter

At the minimum, (height equals diameter):

This holds for any fixed volume — the optimal ratio is a geometric property, not specific to 400 cm³.

Optimization Check-In: Working at r = 5

A can must hold 1000 cm³ of paint.

At cm:

a) What is ?
b) What is SA?

Set up and compute before advancing.

Answer: Can Dimensions at r = 5 cm

Given: cm³, cm

Box Optimization: Setting Up the Equations

Open-top square-base box, cm³. Minimize total SA.

Box Optimization: Table Finds the Minimum

Minimum near cm — optimal height .

Why Real Cans Differ from Optimal Cans

The mathematical optimum gives — but most soup cans are taller and thinner:

• Shelf: tall cans fit more units per row
• Label: more height = more branding space
• Stacking: certain proportions are more stable

The model doesn't include every constraint — always interpret.

Key Takeaways from Deck One

✓ Modeling cycle: Define → Constrain → Set Up → Explore → Interpret

✓ Volume constraint → one variable; optimal cylinder

Watch out:

• Constraint (fixed) vs. objective (minimize) — always label both
• Always interpret: does the solution make physical sense?

Preview: Constraint Design and Grid Systems

Deck 2 builds on today's modeling cycle:

• Chunk 3: Constraint-based design — shipping boxes, water tanks, packaging efficiency
• Chunk 4: Geometry in 2D design — typographic grids, golden ratio, page proportions

Design shapes not only containers but every page and screen.