OptimizationProblems

Lesson 4.7

Applying Our Concepts

• We know aboutmax and min …

• Now how can we use thoseprinciples?

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.

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?

What size squares to maximize the

volume?

30”

60”

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”

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 StationPower Station500

2300

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 StationPower Station500

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

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

Try Animated

Demo

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

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 Ball

• Find T'(x)

• Set equal to zero

• Find optimum y

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 Ball

• r = 6.4 m/s

• s = .910 m/s

Elvis Fetches the Tennis Ball

• Results of trials

Elvis Fetches the Tennis Ball

• Results graphed

Elvis Fetches the Tennis Ball

• With graph of optimum function

Assignment B

• Lesson 4.7 B

• Page 268

• Exercises 43, 47, 54, 55, 58, 59, 60