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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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applying our concepts
Applying Our Concepts
  • We know aboutmax and min …
  • Now how can we use thoseprinciples?
optimization strategy
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”
example maximize volume
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

use the strategy
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”

minimize cost
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

use the strategy7

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

optimizing an angle of observation
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

?

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assignment a
Assignment A
  • Lesson 4.7A
  • Page 265
  • Exercises 1 – 35 odd

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elvis fetches the tennis ball
Elvis Fetches the Tennis Ball
  • Let r be therunning velocity
  • Let s be theswimming velocity
  • Find equation ofTime as function of y

z

elvis fetches the tennis ball11
Elvis Fetches the Tennis Ball
  • Find T'(x)
  • Set equal to zero
  • Find optimum y
elvis fetches the tennis ball12
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
elvis fetches the tennis ball13
Elvis Fetches the Tennis Ball
  • r = 6.4 m/s
  • s = .910 m/s
elvis fetches the tennis ball16
Elvis Fetches the Tennis Ball
  • With graph of optimum function
assignment b
Assignment B
  • Lesson 4.7 B
  • Page 268
  • Exercises 43, 47, 54, 55, 58, 59, 60