1. 3.7: Optimization Homework: p.220 17, 19, 21, 29, 33, 47
Standards
EK2.3C3 – The derivative can be used to solve optimization problems, that is, finding a maximum or minimum value of a function over a given interval
Learning Objectives: Solve applied minimum and maximum problems

2. Does 𝑽= 𝒔 𝟑 or does 𝑽= 𝒔 𝟐 ⋅𝒉 or does 𝑽=𝒍⋅𝒘⋅𝒉?
Concept 1: Formulas
Does 𝑽= 𝒔 𝟑 or does 𝑽= 𝒔 𝟐 ⋅𝒉 or does 𝑽=𝒍⋅𝒘⋅𝒉?

3. The Equation/Formula that is to be used in the optimization
Concept 1: Vocabulary
Primary Equation
The Equation/Formula that is to be used in the optimization

4. Teacher Example 1: Finding Maximum Volume
An open top box with a square base needs to have a surface area of 108 cubic inches. How large would the sides have to be to maximize the volume?

5. Teacher Example 1: Finding Maximum Volume
Step 1: Volume Formula Step 2: Surface Area Step 3: Single Variable Conversion Step 4: Feasible Region Step 5: Maximize Step 6: Solve for the variable Step 7: Evaluate

6. Teacher Example 1: Finding Maximum Volume

7. Concept 1 (Continued): Guidelines for Solving Optimization Problems
On page 216, bottom of the page, enter the appropriate information into your notes. 2 minutes + 10 minutes for SLE

8. Student Led Example 1: Finding Maximum Volume
Handout: Problem 1

9. Student Led Example 1: Handout – Problem 1
This image is here to ASSIST and has nothing to do with the actual problem…they are, however, ridiculously similar.

10. Student Led Example 1: Finding Maximum Volume

11. Student Led Example 1: Finding Maximum Volume
𝑽= 𝟑𝟎−𝟐𝒙 𝟏𝟒−𝟐𝒙 ⋅𝒙
𝑽 ′ =𝟏𝟐 𝒙 𝟐 −𝟏𝟕𝟔𝒙+𝟒𝟐𝟎
=𝟒(𝟑𝒙−𝟑𝟓)(𝒙−𝟑)
10 Minutes

12. Student Led Example 1: Finding Maximum Volume
𝟎≤𝒙≤𝟕
Feasible Region:
Critical Points:
𝒙=𝟑

13. Teacher Example 2: Minimum Distance
Which points on the graph 𝒚=𝟒− 𝒙 𝟐 are closest to the point 𝟎,𝟐 ?

14. Teacher Example 2: Minimum Distance
Which points on the graph 𝒚=𝟒− 𝒙 𝟐 are closest to the point 𝟎,𝟐 ?

15. Teacher Example 2: Minimum Distance
Which points on the graph 𝒚=𝟒− 𝒙 𝟐 are closest to the point 𝟎,𝟐 ? What quantity needs to be minimized? Or, in other words, what is the PRIMARY EQUATION?

16. Teacher Example 2: Minimum Distance
Which points on the graph 𝒚=𝟒− 𝒙 𝟐 are closest to the point 𝟎,𝟐 ?
𝒅= 𝒙−𝟎 𝟐 + 𝒚−𝟐 𝟐
𝒚=𝟒− 𝒙 𝟐
𝒅= 𝒙 𝟐 + 𝟒− 𝒙 𝟐 −𝟐 𝟐
𝒅= 𝒙 𝟒 −𝟑 𝒙 𝟐 +𝟒

17. Teacher Example 2: Minimum Distance
Which points on the graph 𝒚=𝟒− 𝒙 𝟐 are closest to the point 𝟎,𝟐 ?
𝒅 is smallest when the radicand is smallest.
𝒙 𝟒 −𝟑 𝒙 𝟐 +𝟒 ′ =𝟒 𝒙 𝟑 −𝟔𝒙
=𝟐𝒙 𝒙 𝟐 −𝟑
Critical values occur at 𝒙= 𝟎,± 𝟑 𝟐

18. Teacher Example 2: Minimum Distance
Which points on the graph 𝒚=𝟒− 𝒙 𝟐 are closest to the point 𝟎,𝟐 ?

19. Teacher Example 2: Minimum Distance
Which points on the graph 𝒚=𝟒− 𝒙 𝟐 are closest to the point 𝟎,𝟐 ? 𝒇 𝟑 𝟐 = 𝟓 𝟐 𝒇 − 𝟑 𝟐 = 𝟓 𝟐

20. Student Led Example 2: Finding Minimum Distance
Handout: Problem 2

21. Student Led Example 2: Finding Minimum Distance
Function to be minimized:
𝒅= 𝒙−𝟕 𝟐 + 𝒙 𝟐
Critical Points:
𝒙= 𝟏𝟑 𝟐
Feasible Region:
−∞<𝒙<∞

22. Teacher Example 3: Finding Minimum Length
Two posts, one 12 feet high and the other 28 feet high, stand 30 feet apart. They are to be stayed by two wires, attached to a single stake, running from ground level to the top of each post. Where should the
stake be placed to use the least
amount of wire?

23. Teacher Example 3: Finding Minimum Length
Pictures! 

24. Teacher Example 3: Finding Minimum Length
Function to be Minimized:
𝑾=𝒚+𝒛

25. Teacher Example 3: Finding Minimum Length
Function to be Minimized: 𝑾=𝒚+𝒛
𝒚= 𝒙 𝟐 +𝟏𝟒𝟒
𝒛= 𝟑𝟎−𝒙 𝟐 + 𝟐 𝟖 𝟐 = 𝒙 𝟐 −𝟔𝟎𝒙+𝟏𝟔𝟖𝟒

26. Teacher Example 3: Finding Minimum Length
Function to be Minimized: 𝑾=𝒚+𝒛
𝑊 ′ = 𝑦 ′ +𝑧′
𝑧 ′ = 𝑥−30 𝑥 2 −60𝑥+1684
𝑦 ′ = 𝑥 𝑥
𝑑𝑊 𝑑𝑥 = 𝑥 𝑥 𝑥−30 𝑥 2 −60𝑥+1684

27. Teacher Example 3: Finding Minimum Length
Function to be Minimized: 𝑾=𝒚+𝒛
𝑥 𝑥 𝑥−30 𝑥 2 −60𝑥 =0
After some ridiculous algebra we get…
𝑥={9,−22.5}

28. Teacher Example 3: Finding Minimum Length
Function to be Minimized: 𝑾=𝒚+𝒛
𝑥= 9,−22.5
𝑊 𝑥 =𝑦 𝑥 +𝑧 𝑥
𝑦 9 = = = =15
𝑧 𝑥 = − = 81− = =35
𝑊 9 =50

29. Student Led Example 3: Finding Minimum Length
Handout: Problem 3

30. Exit Task – Complete if not done
Enter into your notes: “Guidelines for Solving Applied Minimum and Maximum problems” on page 216