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Optimizing Functions: Maximizing Volume, Profit, and Area in Real-World Problems

This lesson focuses on the optimization of continuous functions by maximizing or minimizing specific quantities. Students will explore practical examples, such as maximizing the volume of an open-top box made from a rectangular sheet of tin and optimizing the area of a pigpen using limited fencing. The session includes the formulation of mathematical models, identifying critical points, and interpreting solutions based on original problems. Assignments and exercises from sections 4.4 and 4.5 will deepen understanding of profit maximization and cost minimization.

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Optimizing Functions: Maximizing Volume, Profit, and Area in Real-World Problems

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  1. Warm-Up: December 17, 2012 • Draw a graph of a continuous function that satisfies all of the following criteria:

  2. Homework Questions?

  3. Modeling and Optimization Section 4.4

  4. Optimization • To optimize something means to maximize or minimize some aspect of it. • Examples: • Maximize profit • Maximize area • Maximize volume • Minimize cost • Minimize material used to produce a good

  5. Solving the Problems • Write an equation of a single variable that represents the quantity you are trying to optimize. • Graph the function and determine the domain. • Find the critical points and endpoints. • Solve the mathematical model. • Interpret the solution in terms of the original problem.

  6. Example 1 • An open-top box is to be made by cutting congruent squares from each corner of a 30” by 40” sheet of tin and bending up the sides. • How large should the squares be to make the box hold as much as possible? • What is the resulting volume?

  7. Example 1 • Maximize volume of box from 30” by 40” sheet

  8. Maximum Profit • Maximum profit (if it exists) occurs at a production level at which marginal revenue equals marginal cost.

  9. Example 2 • Suppose r(x) represents revenue, and c(x) represents cost to produce x thousands of units. Is there a production level that maximizes profit? If so, what is it?

  10. Example 2 – Maximize Profit

  11. Assignment • Read Section 4.4 (pages 206-213) • Answer any 10 Exercises on pages 214-220 • Read Section 4.5 (pages 220-228)

  12. Warm-Up: December 18, 2012 • Farmer John wants to build a rectangular pigpen on the side of the barn. He has 30 m of fence. If he uses the side of the barn as one side of the pigpen, how should he build the pigpen in order to maximize its area?

  13. Homework Questions?

  14. Minimizing Average Cost • The production level at which average cost is smallest (if such level exists) is a level at which the average cost equals the marginal cost.

  15. Example 3 • Suppose c(x) represents cost to produce x thousands of units. Is there a production level that minimizes average cost? If so, what is that production level?

  16. Example 3 – Minimize Cost

  17. Assignment • Read Section 4.4 (pages 206-213) • Answer any 10 Exercises on pages 214-220 • Read Section 4.5 (pages 220-228) • Quiz on 4.1-4.3 Tomorrow

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