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Setting Up Optimization Problems

From Words to Calculus

A farmer has fencing. A company wants to maximize profit. An engineer needs to minimize material costs. These real-world problems share a common structure: something needs to be made as large or as small as possible, subject to constraints.

The power of calculus is that derivatives tell us where functions reach their extreme values. But before you can use derivatives, you must translate the English description into a mathematical function. This translation step—setting up the problem—is often harder than the calculus that follows.

The good news: there's a systematic procedure that works for nearly every optimization problem.

Prerequisite Map

Quick Reference

The Six-Step Procedure

Step 1: Understand the Problem

Read carefully. Ask yourself:

• What quantity do I want to maximize or minimize?
• What quantities are given?
• What constraints limit my choices?

Common words that signal optimization:

• Maximum/minimum
• Largest/smallest
• Greatest/least
• Optimal/best

Step 2: Draw a Diagram

Most optimization problems have a geometric component. A clear diagram:

• Makes relationships visible
• Prevents errors in equations
• Reveals what variables you need

Example: Rectangular field along a river

          ┌─────────────────────────────┐
          │                             │
      x   │        Field                │  x
          │                             │
          └─────────────────────────────┘
    ══════════════════════════════════════════
                    RIVER
                    (no fence needed)
                        y

Step 3: Introduce Notation

Assign symbols to:

• The quantity to optimize (call it $Q$)
• The unknown dimensions or quantities

Use suggestive letters: $A$ for area, $V$ for volume, $t$ for time, $d$ for distance.

Step 4: Write the Objective Function

Express the quantity to optimize in terms of your variables:

$$Q = \text{(expression in terms of variables)}$$

This is your objective function—the function you'll eventually maximize or minimize.

Step 5: Reduce to One Variable

Here's where constraints become crucial. If your objective function has multiple variables, use the given information to eliminate all but one.

Example: If $Q = xy$ and you know $2x + y = 100$, then:

• Solve the constraint: $y = 100 - 2x$
• Substitute: $Q(x) = x(100 - 2x) = 100x - 2x^2$

Now $Q$ is a function of $x$ alone, ready for calculus.

Don't forget the domain: What values can your variable actually take? The domain comes from physical constraints (lengths must be positive, angles between 0 and $\pi$, etc.).

Step 6: Solve Using Calculus

This is covered in the next skill: Optimization Solution Methods.

Worked Example: The Fencing Problem

Problem: A farmer has 2400 ft of fencing to enclose a rectangular field along a straight river. No fence is needed along the river. What dimensions maximize the enclosed area?

Step 1: Understand

• Maximize: area of rectangle
• Given: 2400 ft of fencing
• Constraint: no fence on river side

Step 2: Diagram

          ┌─────────────────────────────┐
          │                             │
      x   │        Field                │  x
          │                             │
          └─────────────────────────────┘
    ══════════════════════════════════════════
                    RIVER
                        y

Step 3: Notation

• $A$ = area (what we maximize)
• $x$ = depth (perpendicular to river)
• $y$ = width (parallel to river)

Step 4: Objective Function

$$A = xy$$

Step 5: Reduce to One Variable

The constraint: fencing = two sides of length $x$ + one side of length $y$

$$2x + y = 2400$$

Solve for $y$: $y = 2400 - 2x$

Substitute into objective:

$$A(x) = x(2400 - 2x) = 2400x - 2x^2$$

Domain: $x$ must be positive. Maximum $x$ occurs when all fencing goes to depth: $2x = 2400 \Rightarrow x = 1200$. So the domain is $0 \le x \le 1200$.

The setup is complete. We have:

$$\boxed{A(x) = 2400x - 2x^2, \quad 0 \le x \le 1200}$$

Common Constraint Types

Practice Problems

    ┌────┬────┬────┐
    │    │    │    │
  y │    │    │    │
    │    │    │    │
    └────┴────┴────┘
           x

    ┌──┬──────────────────┬──┐
    │x │                  │x │
    ├──┼──────────────────┼──┤
    │  │                  │  │
    │  │     30 cm        │  │
    │  │                  │  │
    ├──┼──────────────────┼──┤
    │x │      16 cm       │x │
    └──┴──────────────────┴──┘

CCI-Style Conceptual Questions

Mastery Checklist

• [ ] I can identify what quantity to maximize or minimize from a word problem
• [ ] I can draw a clear diagram with labeled variables
• [ ] I can write the objective function in terms of given variables
• [ ] I can use constraints to reduce to a single-variable function
• [ ] I can determine the domain from physical restrictions
• [ ] I can recognize common constraint types (perimeter, volume, distance, etc.)

Mental Model

The Translation Process:

Think of optimization setup like translating between languages:

The constraint is the bridge between variables. It lets you express everything in terms of one unknown—and that's when calculus can take over.

Connections

Looking back:

• Critical points and the first derivative test (§3.1, §3.3) tell us how to find extrema once we have the function

Looking ahead:

• Optimization Solution Methods covers solving after setup
• Related rates (§3.9) also translates word problems to calculus, but finds rates instead of extrema

Real-world connections:

• Engineers minimize material costs while meeting strength requirements
• Economists maximize profit subject to production constraints
• Nature often "optimizes"—light takes the path of least time (Fermat's principle)

Last updated: 2026-01-22