How can we use derivatives to find maximum or minimum amounts?
Essential Question Essential Question Essential Question Essential Question Essential Question Essential Question Essential Question
Week 7, Lesson 1
1. Warm Up
2. Notes­ Optimization
3. ICA­ Group Activity
4. Week 7 HW
Optimization
How can we use derivatives to find maximum or minimum amounts?
Standard 7.6
57
Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up
Warm­up: Unit 7c Vocabulary
Optimization: A method using derivatives to find the "optimal" or "best" value
Maximize: The greatest quantity or amount possible
Minimize: The least or smallest amount or quantity possible, attainable, or required.
Volume: The amount of space, measured in cubic units, that an object or substance occupies.
Surface Area: The total area of the surface of a three­dimensional object.
Area: The amount of space inside the boundary of a flat (2­dimensional) object
Perimeter: The distance around a two­dimensional shape.
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Standard 7.6
Optimization
Optimization: ­A method using derivatives to find the "optimal" or "best" value
This is used to make sure that you do not use too much or too little of material
i.e.­ Cereal boxes don't want to use more cardboard than they need
Method:
­Identify the variables
­Identify the formulas
­Identify the "requirements"
­Solve the "requirement" for one variable
­Plug in to the other formula and find the maximum/minimum
Example 1 What is the largest a rectangle's area can be if the width and length add up to 50 mm?
Important Variables
Information
Area = A
width = w
length = L
Formulas
Requirements
A = Lw
L + w
L + w = 50
width of Example 2 A rectangle's perimeter is 400 cm. What is the length and the side lengths that will produce the largest area?
Variables
Perimeter= P
width =w
length= L
area = A
Summary:
Formulas
Requirements
A=Lw
400= 2L+2w
P= 2L+2w
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Group Activity
Techniques for solving optimization problems: 1. Draw a picture of the situation 2. Identify the quantity you want to optimize 3. Find a mathematical expression for the quantity that you want to optimize 4. Eliminate extra variables (there should only be one variable in the expression) 5. Find the domain of your function 6. Find global max and min Use the techniques together to solve the problems below as a group. The group to solve all problems first will get extra credit. 1. A farmer is building a rectangular pen for his pigs. He has exactly 100 ft of fencing material. (b) Suppose the farmer finds that he can build his pen right next to his neighbour’s property so that he can use the neighbours fence for one side of the pen (see picture below). Find the dimensions of the pen that he can build to maximizes the area enclosed by the fence. 2. Suppose you had 102 m of fencing to make two side­by­
side enclosures as shown. What is the maximum area that you could enclose? Closure Closure Closure Closure Closure Closure Closure Closure Closure Closure Closure Closure Closure Closure Closure Closure Closure Closure Closure
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Right Side...
Write a summary that answers the essential question.
Left Side...
Quick write: Describe in your own words what are stationary points.
How can we use derivatives to find maximum or minimum amounts?
Essential Question Essential Question Essential Question Essential Question Essential Question Essential Question Essential Question
Week 7, Lesson 2
1. Warm­up
2. Notes­ Optimization Day 2
3. ICA­ Farmer's Fences
How can we use derivatives to find maximum or minimum amounts?
Optimization Day 2
Standards 7.6
59
Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up
Warm­up: At what value of x does the function g(x) reach its minimum?
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Standards 7.6
Optimization Day 2
Practice A farmer needs to build a rectangular field of fence. Problem #1: He needs to have an area of 50,000 sq. ft. What is the minimum amount of perimeter needed?
Variables Formulas Requirements
A=Lw
P=2L+2w
Area= A
Perimeter= P
Length=L
Width=w
50000=Lw
Farmer Henderson is installing fence along the creek. Practice Each equal section of fencing is for 4 different types of animals. Problem #2:
He has 1200 meters of fencing. What is the largest area he can fence off for the cows?
W
Cows
W
Sheep
W
Pigs
W
Horses W
L
Variables Formulas Requirements
A= Area
P=perimeter
y= width x= the whole length
Summary:
A=xy
5y+x
5y+x=1200
Page 58
Example 1 What is the largest a rectangle's area can be if the width and length add up to 50 mm?
Variables
Formulas
Area = A
A = Lw
L + w
width = w
length = L
Solving for 1 Requirement
L + w = 50
Requirements
L + w = 50
Plug into other formula and find max/min
A = Lw
A = (50­w)w
A = (50w­w2)
0 = (50­2w)
A rectangle's perimeter is 400 cm. What is the length and width of Example 2 the side lengths that will produce the largest area?
Variables
Perimeter= P
width =w
Formulas
Requirements
A=Lw
400= 2L+2w
P= 2L+2w
length= L
area = A
Solving for 1 Requirement
400= 2L+2w
Plug into other formula and find max/min
A=Lw
A=(200­w)w
A=200w-w2
A'=0=200-2w
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Practice Problems
1. Four pens will be built side by side along a river by using 150 feet of fencing. What dimensions will maximize the area of the pens. w
Cows
w
Sheep
w
Pigs
w
Horses w
L
Variables
A= Area
P=perimeter
w= width L= the whole length
Solving for 1 Requirement
5w+L=150
Formulas
Requirements
A=wL
5w+L=150
5w+L
Plug into other formula and find max/min A=Lw
A=(150­5w)w
A=(150w­5w2)
A'=0=(150­10w)
2. Suppose you had to use exactly 200 m of fencing to make either one square enclosure or two separate square enclosures One Square Enclosure Two separate Enclosures of any size you wished. What plan would give you the least area? What plan would give you the greatest are? 3. Suppose a farmer has to build his pig pen in an triangular area bounded by a 12 ft stretch of river, a 9 ft stretch of wall and a 15 ft stretch of forrest (see picture below). Find the dimensions of the pen that he can build to maximizes the area enclosed by the fence Closure Closure Closure Closure Closure Closure Closure Closure Closure Closure Closure Closure Closure Closure Closure Closure Closure Closure Closure
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Right Side...
Write a summary that answers the essential question.
Left Side...
Quick write: Describe in your own words the purpose of setting your derivative equal to zero.
HOW CAN OPTIMIZATION RELATE TO 3D SHAPES?
Essential Question Essential Question Essential Question Essential Question Essential Question Essential Question Essential Question
Week 7, Lesson 3
1. Warm Up
2. Notes­ Maximizing Volume
3. ICA­ Paper Boxes
Maximizing Volume
HOW CAN OPTIMIZATION RELATE TO 3D SHAPES?
Standard 7.6
61
Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up Warm­up
Warm­up: Find the volume of the rectangular prisms below. x
inches
12cm
8cm
20cm
V= Lwh
7 ­ 2x
inches
8 ­ 3x
inches
V= Lwh
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Standard 7.6
Maximizing Volume
Given: You are given a rectangular piece of paper and a pair of scissors. How can you turn this into a box (missing the top) with the greatest volume?
w
?
l
­You must make 4 equal square cuts in the corners
Creating the Box: Quadratic
Formula:
c
­These cuts are length c
c
c
c
c
c
c
­Length:
l­2c
­Width:
w­2c
­Height:
c
­Volume:
(l­2c)(w­2c)(c)
When setting the derivative = 0 and solving, you will almost NEVER be able to factor. ax2+bx+c
Solutions:
?
End with this...
Start with this...
c
?
Check your answers...
Do they make sense?!
Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes Notes
Maximizing Volume
Standard 7.6
Example 1­
A marble company has 8in by 8in pieces of cardboard to turn in to boxes to ship marbles. What is the largest volume they can create?
8in
8in
8­2x
8­2x
x
Example 2­ A textbook company has 25in by 30in pieces of cardboard to turn in to boxes to ship their books. What is the largest volume they can create?
30in
25in
25­2x
30­2x
x
Summary:
ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity
ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity ICA: In Class Activity
Paper Boxes
A standard piece of paper is 8.5 in x 11 in
Create a box with maximum volume
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Right Side...
Write a summary that answers the essential question.
Left Side...
Quick write: Explain why it is critical to draw a picture of the situation before you start any problem.
Homework Week 7
1. An apple orchard wants to plant more trees. There are already 50 trees in the orchard, and each tree produces 800 apples. For each new tree planted, all trees will produce 10
less apples. How many new trees should be added to maximize the amount of apples produced?
2. At a business, each employee profits the company $1000 per week. They have 300 employees. If they lay off 1 employee, every other employee will profit an additional $10 each week. How many employees should the company lay off in order to maximize their profits?
3. Ms. Carroll wants to remodel her backyard. She wants to have 800 sq ft of grass with a rectangular brick border, but brick is expensive. What is the minimum amount of brick she can purchase while keeping 800 sq ft of grass?
4. Mrs. Sonnier wants to plant 3 seperate gardens side by side along a brick wall. (pictured below) She has 60 feet of garden fence. What is the overall maximum area she can have for her garden?
Brick Wall
w
L
5. Mr. Henderson is looking for two positive numbers a and b that add up to 9. What are the optimal numbers for a and b to find the maximum possibility of Attachments
W2L1.docx
Project Survey.docx
Project Survey.xlsx
Project Questionnaire.docx
Project Questionnaire.xlsx
W5L3.docx