1. Optimization Problems Lesson 4.7

2. Applying Our Concepts We know about max and min … Now how can we use those principles?

3. Optimization Strategy 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” Note Guidelines, pg 260 from text. Note Guidelines, pg 260 from text.

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

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 x 30” 60”

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. Use the Strategy 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 View Spreadsheet Model View Spreadsheet Model View example of a dog who seemed to know this principle View example of a dog who seemed to know this principle

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 wall should the camera be placed to photograph the mural with the Largest possible angle? ? 8 13 4 Try Animated Demo Try Animated Demo

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

10. Elvis Fetches the Tennis Ball Let r be the running velocity Let s be the swimming velocity Find equation of Time 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