L1 Numbers Problems and Geometric Applications - Part 5
Completion requirements
Unit 5
Applications of Derivatives
A. Maximum and Minimum Problems
Lesson 1: Numbers Problems and Geometric Applications
When solving problems involving extreme values, the greatest challenge is often in expressing the information in the problem using algebra. To solve these problems, a function for the value to be minimized or maximized must first be set up. As seen
in the video, four steps were provided to solve problems. The following list is an expanded version of the four steps from the video.
Steps to Solving Applied Maximum and Minimum Problems
Steps to Solving Applied Maximum and Minimum Problems
- Read the problem carefully – more than once through is often necessary. From the description of the problem, identify the unknown(s), the given quantities/values, and determine what is to be maximized or minimized.
- If possible, draw a diagram and label it with what is given and what is needed.
- Select variables to represent the unknowns and the quantity to be maximized or minimized. Write statements to define the variables.
- Write an equation for the function expressing the quantity to be minimized or maximized. Express one variable in terms of the other, then rewrite the function to be maximized or minimized in terms of a single variable.
- Take the derivative of the function, set it equal to zero, and solve for the variable to find the value(s) where the maximum or minimum of the function occurs. Determine if the solution(s) is/are part of the domain of the function.
- Use the result(s) from Step 5 to determine the values of the other unknowns using substitution.
- Verify the solution corresponds to a maximum or minimum value using an appropriate test such as the second derivative test.
- Write a concluding statement, ensuring all questions posed have been answered.
Watch the video Steps to Solving Maximum and Minimum Problems Continued for an additional example on optimizing materials.
A rectangular page is to contain «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»150«/mn»«mo»§#160;«/mo»«msup»«mi»cm«/mi»«mn»2«/mn»«/msup»«/mstyle»«/math» of printing. The margins
at the top and bottom of the page are each «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»3«/mn»«mo»§#160;«/mo»«mi»cm«/mi»«/mrow»«/mstyle»«/math». The margins on each side are
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«mo»§#160;«/mo»«mi»cm«/mi»«/mstyle»«/math». What should be the dimensions of the page if the minimum amount of paper is used?
Draw a diagram and label it with the measurements provided.
Let «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» be the width of the printed area and let «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math» be the length of the printed area.
Therefore, let «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»4«/mn»«/mrow»«/mstyle»«/math» be the width of the page, and let «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«mo»+«/mo»«mn»6«/mn»«/mstyle»«/math» be the length of the page.
Let «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»A«/mi»«/mstyle»«/math» be the area to be minimized.
Solving «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»150«/mn»«mo»=«/mo»«mi»x«/mi»«mi»y«/mi»«/mrow»«/mstyle»«/math» for «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math» gives «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mfrac»«mn»150«/mn»«mi»x«/mi»«/mfrac»«/mrow»«/mstyle»«/math».
Substitute «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mfrac»«mn»150«/mn»«mi»x«/mi»«/mfrac»«/mrow»«/mstyle»«/math» into «math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»A«/mi»«mo»=«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»+«/mo»«mn»4«/mn»«mi»)(«/mi»«mi»y«/mi»«mo»+«/mo»«mn»6«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math».
To find the minimum area, determine the derivative of the function and equate it to zero.
Solve for «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math» where «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»10«/mn»«/mrow»«/mstyle»«/math».
By the second derivative test, the area function is at a minimum where «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»10«/mn»«/mrow»«/mstyle»«/math» because the second derivative is positive at that point (concave up).
When «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»10«/mn»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«mo»=«/mo»«mn»15«/mn»«/mstyle»«/math», the dimensions of the printed page are as follows.
The dimensions of the page should be «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»14«/mn»«mo»§#160;«/mo»«mi»cm«/mi»«/mrow»«/mstyle»«/math» by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»21«/mn»«mo»§#160;«/mo»«mi»cm«/mi»«/mstyle»«/math» to ensure a minimum amount of paper is used.
Let «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» be the width of the printed area and let «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math» be the length of the printed area.
Therefore, let «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»4«/mn»«/mrow»«/mstyle»«/math» be the width of the page, and let «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«mo»+«/mo»«mn»6«/mn»«/mstyle»«/math» be the length of the page.
Let «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»A«/mi»«/mstyle»«/math» be the area to be minimized.
Area of printing on page:
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«msub»«mi»A«/mi»«mi»print«/mi»«/msub»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»x«/mi»«mi»y«/mi»«/mtd»«/mtr»«mtr»«mtd»«mn»150«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»x«/mi»«mi»y«/mi»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Area of printed page with required margins:
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»A«/mi»«mo»=«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»+«/mo»«mn»4«/mn»«mi»)(«/mi»«mi»y«/mi»«mo»+«/mo»«mn»6«/mn»«mi
mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math»
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»A«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»+«/mo»«mn»4«/mn»«mi
mathvariant=¨normal¨»)«/mi»«mfenced»«mrow»«mfrac»«mn»150«/mn»«mi»x«/mi»«/mfrac»«mo»+«/mo»«mn»6«/mn»«/mrow»«/mfenced»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»150«/mn»«mo»+«/mo»«mn»6«/mn»«mi»x«/mi»«mo»+«/mo»«mfrac»«mn»600«/mn»«mi»x«/mi»«/mfrac»«mo»+«/mo»«mn»24«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»174«/mn»«mo»+«/mo»«mn»6«/mn»«mi»x«/mi»«mo»+«/mo»«mfrac»«mn»600«/mn»«mi»x«/mi»«/mfrac»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
To find the minimum area, determine the derivative of the function and equate it to zero.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»A«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»174«/mn»«mo»+«/mo»«mn»6«/mn»«mi»x«/mi»«mo»+«/mo»«mn»600«/mn»«msup»«mi»x«/mi»«mrow»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«/msup»«/mtd»«/mtr»«mtr»«mtd»«mfrac»«mrow»«mi»d«/mi»«mi»A«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»6«/mn»«mo»§#8722;«/mo»«mn»600«/mn»«msup»«mi»x«/mi»«mrow»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/msup»«/mtd»«/mtr»«mtr»«mtd/»«mtd/»«mtd/»«/mtr»«mtr»«mtd»«mn»0«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»6«/mn»«mo»§#8722;«/mo»«mfrac»«mn»600«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mfrac»«mn»600«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»6«/mn»«/mtd»«/mtr»«mtr»«mtd»«mn»600«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»6«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mtd»«/mtr»«mtr»«mtd»«mn»100«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mtd»«/mtr»«mtr»«mtd»«mn»10«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»x«/mi»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
| If «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»=«/mo»«mn»100«/mn»«/mrow»«/mstyle»«/math», «math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mo»§#177;«/mo»«mn»10«/mn»«/mrow»«/mstyle»«/math». But, since «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» is a length, it cannot be negative.
|
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»y«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»150«/mn»«mi»x«/mi»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»15«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfrac»«mrow»«msup»«mi»d«/mi»«mn»2«/mn»«/msup»«mi»A«/mi»«/mrow»«mrow»«mi»d«/mi»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«mo»=«/mo»«mfrac»«mrow»«mn»1«/mn»«mo»§#160;«/mo»«mn»200«/mn»«/mrow»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«/mfrac»«/mrow»«/mstyle»«/math»
By the second derivative test, the area function is at a minimum where «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»10«/mn»«/mrow»«/mstyle»«/math» because the second derivative is positive at that point (concave up).
When «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»10«/mn»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«mo»=«/mo»«mn»15«/mn»«/mstyle»«/math», the dimensions of the printed page are as follows.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left center center right center left¨»«mtr»«mtd»«mi»width«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»x«/mi»«mo»+«/mo»«mn»4«/mn»«/mtd»«mtd/»«mtd/»«mtd»«mi»length«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»y«/mi»«mo»+«/mo»«mn»6«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»10«/mn»«mo»+«/mo»«mn»4«/mn»«/mtd»«mtd/»«mtd/»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»15«/mn»«mo»+«/mo»«mn»6«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»14«/mn»«/mtd»«mtd/»«mtd/»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»21«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
The dimensions of the page should be «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»14«/mn»«mo»§#160;«/mo»«mi»cm«/mi»«/mrow»«/mstyle»«/math» by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»21«/mn»«mo»§#160;«/mo»«mi»cm«/mi»«/mstyle»«/math» to ensure a minimum amount of paper is used.