L2 Extreme Values of Distance and Time and Economics - Part 2
Completion requirements
Unit 5
Applications of Derivatives
A. Maximum and Minimum Problems
Lesson 2: Extreme Values of Distance and Time and Economics
In Lesson 1, steps were formulated to solve applied maximum and minimum problems. Those same steps will be applied throughout this lesson.
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.
Suppose an airplane flying west at «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»200«/mn»«mo»§#160;«/mo»«mi»km«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»h«/mi»«/math» passes over a house «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»10«/mn»«/math» minutes before another plane, flying south at «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»300«/mn»«mo»§#160;«/mo»«mi»km«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»h«/mi»«/math», passes over the same house. Assuming both planes are at the same altitude, at what time is the distance between them a minimum? What is that minimum distance?
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«/math» be the position of the first plane and let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«/math» be the position of the second plane.
Initially, the first plane is at the origin «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfenced»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»0«/mn»«/mrow»«/mfenced»«/math». The second plane, travelling at a speed of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»300«/mn»«mo»§#160;«/mo»«mi»km«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»h«/mi»«/math» travels a distance of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mn»300«/mn»«mfenced»«mfrac»«mn»10«/mn»«mn»60«/mn»«/mfrac»«/mfenced»«mo»=«/mo»«mn»50«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi»km«/mi»«/math» in «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»10«/mn»«/math» minutes. Thus, the second plane is initially «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»50«/mn»«mo»§#160;«/mo»«mi»km«/mi»«/math» north of the house.
The initial situation can be shown as follows.
After a period of time «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«/math», in hours, the first plane travels a distance of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»200«/mn»«mi»t«/mi»«mo»§#160;«/mo»«mi»km«/mi»«/math», and the second plane travels «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»300«/mn»«mi»t«/mi»«mo»§#160;«/mo»«mi»km«/mi»«/math». The new positions of the airplanes are shown in the diagram below.
The distance «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»z«/mi»«/math» between the two planes is given by «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup»«mi»z«/mi»«mn»2«/mn»«/msup»«mo»=«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mi»y«/mi»«mn»2«/mn»«/msup»«/math», where «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»=«/mo»«mn»200«/mn»«mi»t«/mi»«/math» and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mn»50«/mn»«mo»§#8722;«/mo»«mn»300«/mn»«mi»t«/mi»«/math».
The distance between the two planes will be a minimum when «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»d«/mi»«mi»z«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mn»0«/mn»«/math». That is, take the derivative of the function with respect to the time variable, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«/math».
At «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»3«/mn»«mn»26«/mn»«/mfrac»«mo»§#160;«/mo»«mi»h«/mi»«/math» after the first plane passes over the house, the distance between the planes will be a minimum.
The minimum distance between the planes is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mn»100«/mn»«msqrt»«mn»13«/mn»«/msqrt»«/mrow»«mn»13«/mn»«/mfrac»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi»km«/mi»«/math».
Initially, the first plane is at the origin «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfenced»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»0«/mn»«/mrow»«/mfenced»«/math». The second plane, travelling at a speed of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»300«/mn»«mo»§#160;«/mo»«mi»km«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»h«/mi»«/math» travels a distance of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mn»300«/mn»«mfenced»«mfrac»«mn»10«/mn»«mn»60«/mn»«/mfrac»«/mfenced»«mo»=«/mo»«mn»50«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi»km«/mi»«/math» in «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»10«/mn»«/math» minutes. Thus, the second plane is initially «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»50«/mn»«mo»§#160;«/mo»«mi»km«/mi»«/math» north of the house.
The initial situation can be shown as follows.
After a period of time «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«/math», in hours, the first plane travels a distance of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»200«/mn»«mi»t«/mi»«mo»§#160;«/mo»«mi»km«/mi»«/math», and the second plane travels «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»300«/mn»«mi»t«/mi»«mo»§#160;«/mo»«mi»km«/mi»«/math». The new positions of the airplanes are shown in the diagram below.
The distance «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»z«/mi»«/math» between the two planes is given by «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup»«mi»z«/mi»«mn»2«/mn»«/msup»«mo»=«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mi»y«/mi»«mn»2«/mn»«/msup»«/math», where «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»=«/mo»«mn»200«/mn»«mi»t«/mi»«/math» and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mn»50«/mn»«mo»§#8722;«/mo»«mn»300«/mn»«mi»t«/mi»«/math».
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup»«mi»z«/mi»«mn»2«/mn»«/msup»«mo»=«/mo»«msup»«mfenced»«mrow»«mn»200«/mn»«mi»t«/mi»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mfenced»«mrow»«mn»50«/mn»«mo»§#8722;«/mo»«mn»300«/mn»«mi»t«/mi»«/mrow»«/mfenced»«mn»2«/mn»«/msup»«/math»
The distance between the two planes will be a minimum when «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»d«/mi»«mi»z«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mn»0«/mn»«/math». That is, take the derivative of the function with respect to the time variable, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«/math».
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mn»2«/mn»«mi»z«/mi»«mo»§#8729;«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»z«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»2«/mn»«mfenced»«mrow»«mn»200«/mn»«mi»t«/mi»«/mrow»«/mfenced»«mo»§#8729;«/mo»«mn»200«/mn»«mo»+«/mo»«mn»2«/mn»«mfenced»«mrow»«mn»50«/mn»«mo»§#8722;«/mo»«mn»300«/mn»«mi»t«/mi»«/mrow»«/mfenced»«mfenced»«mrow»«mo»§#8722;«/mo»«mn»300«/mn»«/mrow»«/mfenced»«/mtd»«/mtr»«mtr»«mtd»«mn»2«/mn»«mi»z«/mi»«mo»§#8729;«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»z«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»80«/mn»«mi mathvariant=¨normal¨» «/mi»«mn»000«/mn»«mi»t«/mi»«mo»§#8722;«/mo»«mn»30«/mn»«mspace width=¨0.125em¨/»«mn»000«/mn»«mo»+«/mo»«mn»180«/mn»«mi mathvariant=¨normal¨» «/mi»«mn»000«/mn»«mi»t«/mi»«/mtd»«/mtr»«mtr»«mtd»«mfrac»«mrow»«mi»d«/mi»«mi»z«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»40«/mn»«mi mathvariant=¨normal¨» «/mi»«mn»000«/mn»«mi»t«/mi»«mo»§#8722;«/mo»«mn»15«/mn»«mspace width=¨0.125em¨/»«mn»000«/mn»«mo»+«/mo»«mn»90«/mn»«mi mathvariant=¨normal¨» «/mi»«mn»000«/mn»«mi»t«/mi»«/mrow»«mi»z«/mi»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»130«/mn»«mi mathvariant=¨normal¨» «/mi»«mn»000«/mn»«mi»t«/mi»«mo»§#8722;«/mo»«mn»15«/mn»«mi mathvariant=¨normal¨» «/mi»«mn»000«/mn»«/mrow»«mi»z«/mi»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd/»«mtd/»«/mtr»«mtr»«mtd»«mn»0«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»130«/mn»«mi mathvariant=¨normal¨» «/mi»«mn»000«/mn»«mi»t«/mi»«mo»§#8722;«/mo»«mn»15«/mn»«mi mathvariant=¨normal¨» «/mi»«mn»000«/mn»«/mrow»«mi»z«/mi»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mn»130«/mn»«mi mathvariant=¨normal¨» «/mi»«mn»000«/mn»«mi»t«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»15«/mn»«mi mathvariant=¨normal¨» «/mi»«mn»000«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»t«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»15«/mn»«mi mathvariant=¨normal¨» «/mi»«mn»000«/mn»«/mrow»«mrow»«mn»130«/mn»«mi mathvariant=¨normal¨» «/mi»«mn»000«/mn»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mi»t«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»3«/mn»«mn»26«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«/math»
At «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»3«/mn»«mn»26«/mn»«/mfrac»«mo»§#160;«/mo»«mi»h«/mi»«/math» after the first plane passes over the house, the distance between the planes will be a minimum.
Solve for «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«/math» where «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«mo»=«/mo»«mfrac»«mn»3«/mn»«mn»26«/mn»«/mfrac»«/math».
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»x«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»200«/mn»«mi»t«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»200«/mn»«mfenced»«mfrac»«mn»3«/mn»«mn»26«/mn»«/mfrac»«/mfenced»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»300«/mn»«mn»13«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«/math»
Solve for «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«/math» where «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«mo»=«/mo»«mfrac»«mn»3«/mn»«mn»26«/mn»«/mfrac»«/math».
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»y«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»50«/mn»«mo»§#8722;«/mo»«mn»300«/mn»«mi»t«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»50«/mn»«mo»§#8722;«/mo»«mn»300«/mn»«mfenced»«mfrac»«mn»3«/mn»«mn»26«/mn»«/mfrac»«/mfenced»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»50«/mn»«mo»§#8722;«/mo»«mfrac»«mn»450«/mn»«mn»13«/mn»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»200«/mn»«mn»13«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«/math»
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«msup»«mi»z«/mi»«mn»2«/mn»«/msup»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mfenced»«mfrac»«mn»300«/mn»«mn»13«/mn»«/mfrac»«/mfenced»«mn»2«/mn»«/msup»«mo»+«/mo»«msup»«mfenced»«mfrac»«mn»200«/mn»«mn»13«/mn»«/mfrac»«/mfenced»«mn»2«/mn»«/msup»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»90«/mn»«mo»§#160;«/mo»«mn»000«/mn»«/mrow»«mn»169«/mn»«/mfrac»«mo»+«/mo»«mfrac»«mrow»«mn»40«/mn»«mo»§#160;«/mo»«mn»000«/mn»«/mrow»«mn»169«/mn»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»130«/mn»«mo»§#160;«/mo»«mn»000«/mn»«/mrow»«mn»169«/mn»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»10«/mn»«mo»§#160;«/mo»«mn»000«/mn»«/mrow»«mn»13«/mn»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mi»z«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msqrt»«mfrac»«mrow»«mn»10«/mn»«mo»§#160;«/mo»«mn»000«/mn»«/mrow»«mn»13«/mn»«/mfrac»«/msqrt»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»100«/mn»«msqrt»«mn»13«/mn»«/msqrt»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»100«/mn»«msqrt»«mn»13«/mn»«/msqrt»«/mrow»«mn»13«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«/math»
The minimum distance between the planes is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mn»100«/mn»«msqrt»«mn»13«/mn»«/msqrt»«/mrow»«mn»13«/mn»«/mfrac»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi»km«/mi»«/math».