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Optimization Problems

Optimization Problems. Lesson 4.7. Applying Our Concepts. We know about max and min … Now how can we use those principles?. Optimization Strategy. Note Guidelines, pg 260 from text. When appropriate, draw a picture Focus on quantity to be optimized

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Optimization Problems

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  1. OptimizationProblems Lesson 4.7

  2. Applying Our Concepts • We know aboutmax and min … • Now how can we use thoseprinciples?

  3. Optimization Strategy Note Guidelines, pg 260 from text. • When appropriate, draw a picture • Focus on quantity to be optimized • Determine formula involving that quantity • Solve for the variable of the quantity to be optimized • Find practical domain for that variable • Use methods of calculus (min/max strategies) to obtain required optimal value • Check if resulting answer “makes sense”

  4. Example: Maximize Volume • Consider construction of open topped box from single piece of cardboard • Cut squares out of corners 60” 30” What size squares to maximize the volume? Small corner squares Large corner squares

  5. Use the Strategy • What is the quantity to be optimized? • The volume • What are the measurements (in terms of x)? • What is the variable which will manipulated to determine the optimum volume? • Now use calculus principles 60” x 30”

  6. Minimize Cost • We are laying cable • Underground costs $10 per ft • Underwater costs $15 per ft • How should we lay the cable to minimize to cost • From the power station to the island Power Station 500 2300

  7. View example of a dog who seemed to know this principle Use the Strategy View Spreadsheet Model • Determine a formula for the cost • $10 * length of land cable + $15 * length of under water cable • Determine a variable to manipulate which determines the cost • What are the dimensions in terms of this x • Use calculus methods to minimize cost Power Station 500 2300

  8. Optimizing an Angle of Observation • Bottom of an 8 ft high mural is 13 ft above ground • Lens of camera is 4 ft above ground • How far from the wallshould the camera be placed to photographthe mural with theLargest possible angle? 8 13 4 ? Try Animated Demo

  9. Assignment A • Lesson 4.7A • Page 265 • Exercises 1 – 35 odd More examples from another teacher's website

  10. Elvis Fetches the Tennis Ball • Let r be therunning velocity • Let s be theswimming velocity • Find equation ofTime as function of y z

  11. Elvis Fetches the Tennis Ball • Find T'(x) • Set equal to zero • Find optimum y

  12. Elvis Fetches the Tennis Ball • Determine Elvis's quickness • Running • Swimming • Average 3 fastest • r = 6.4 m/s • s = .910 m/s • Plug into optimum equation

  13. Elvis Fetches the Tennis Ball • r = 6.4 m/s • s = .910 m/s

  14. Elvis Fetches the Tennis Ball • Results of trials

  15. Elvis Fetches the Tennis Ball • Results graphed

  16. Elvis Fetches the Tennis Ball • With graph of optimum function

  17. Assignment B • Lesson 4.7 B • Page 268 • Exercises 43, 47, 54, 55, 58, 59, 60

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