Optimization
Problems
Lesson 4.7
Applying Our Concepts
• We know about
max and min …
• Now how can
we use those
principles?
Optimization
Note Guidelines, Strategy
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
• Consider construction of open topped box
from single piece of cardboard
 Cut squares out of corners
60”
30”
Small corner
squares
What size squares to
maximize the
volume?
Large corner
squares
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
60”
principles
x
30”
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 Strategy
• Determine a formula for the cost
 $10 * length of land cable +
View Spreadsheet
Model
$15 * length of under water cable
• Determine
a example
variable
to who
manipulate which
View
of a dog
seemed to know this principle
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
• Bottom of an 8 ft high mural is 13 ft above
ground
• Lens of camera is 4 ft above ground
8
• How far from the wall
should the camera be
placed to photograph
the mural with the
Largest possible angle?

13
4
Try Animated
Demo
?
Assignment A
• Lesson 4.7A
• Page 265
• Exercises 1 – 35 odd
More examples from another
teacher's website
Elvis Fetches the Tennis Ball
• Let r be the
running
velocity
• Let s be the
swimming
velocity
z
• Find equation of
Time as function of y
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