AC Applied Optimization

Motivating Questions

• How to develop a function to model a situation for optimal parameters.
• Use calculus to find the desired maximum or minimum.

Section Overview

• Discusses optimization problems where determining the function to be optimized is part of the problem.
• Example: Using a wire to create shapes that maximize area, and optimizing the volume of a box from a piece of cardboard.
• Emphasizes problem-solving with guided steps and then encourages independence.

Preview Activity 3.4.1

• Problem: Maximize the volume of a rectangular parcel with a square end under postal restrictions.
• Steps:
  1. Define variables for dimensions of the parcel.
  2. Identify the quantity to be optimized.
  3. Establish constraints (e.g., girth + length = 108 inches).
  4. Solve constraints for one variable.
  5. Formulate a volume equation as a function of one variable.
  6. Determine appropriate domain.
  7. Use calculus to find absolute maximum volume and corresponding dimensions.

More Applied Optimization Problems

General Steps for Solving Optimization Problems

1. Draw a Picture and Introduce Variables
   • Understand varying quantities and represent with variables.
   • Label diagrams, possibly using multiple diagrams.
2. Identify the Quantity to be Optimized
   • Write equations for the quantity and any interrelated variables.
3. Determine a Function of a Single Variable
   • Use relationships to eliminate variables and form a function.
4. Decide on the Domain
   • Use physical constraints to determine minimum and maximum variable values.
5. Use Calculus to Identify Extremes
   • Find critical numbers.
   • Use derivative tests or evaluate at endpoints for bounded intervals.
6. Ensure the Question is Answered
   • Differentiate between finding maximum values and dimensions that yield them.

Example Activities

• Activity 3.4.2: Minimize the cost of constructing a soup can.

  • Use variables to derive expressions for volume, surface area, and cost.
  • Find cost function and domain.
  • Use calculus to minimize cost and find dimensions.

• Activity 3.4.3: Minimize travel time for a hiker.

  • Define path and speed variables.
  • Optimize turning point for minimal travel time.

• Activity 3.4.4: Maximize area or perimeter of a rectangle inscribed under a curve.

  • Define variables for dimensions and relationships.
  • Use function of single variable to find extremes.

Tips for Applied Optimization

• Familiarity with geometric formulas for perimeter, area, volume, etc.
• Recognize when problems introduce geometric or algebraic constraints.

Summary

• No single algorithm fits all optimization problems.
• Steps include drawing diagrams, identifying relationships, formulating functions, setting domains, and using calculus to find extrema.

Exercises

• Practice problems include maximizing box volume, minimizing costs, and solving geometric problems with specific constraints.
• Encourage application of the steps and concepts discussed in the section.

Conclusion

• Understanding and applying optimization techniques is crucial in solving practical and applied problems effectively.