L2 Extreme Values of Distance and Time and Economics - Part 6
Completion requirements
Unit 5
Applications of Derivatives
A. Maximum and Minimum Problems
Lesson 2: Extreme Values of Distance and Time and Economics
A revenue function, which gives total revenue, is given by the number of units sold multiplied by the selling price per unit. The derivative of the revenue function is called marginal revenue, which is used to calculate maximum profits.
Profits are found by subtracting costs from revenue.
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»Revenue«/mi»«mi mathvariant=¨normal¨» «/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»selling«/mi»«mi mathvariant=¨normal¨» «/mi»«mi»price«/mi»«mi mathvariant=¨normal¨»
«/mi»«mi»per«/mi»«mi mathvariant=¨normal¨» «/mi»«mi»unit«/mi»«mi mathvariant=¨normal¨» «/mi»«mo»§#8729;«/mo»«mi»number«/mi»«mi mathvariant=¨normal¨» «/mi»«mi»of«/mi»«mi mathvariant=¨normal¨» «/mi»«mi»units«/mi»«/mtd»«/mtr»«mtr»«mtd»«mi»R«/mi»«mi
mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»p«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»§#8729;«/mo»«mi»x«/mi»«/mtd»«/mtr»«/mtable»«/math»
Profits are found by subtracting costs from revenue.
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»Profit«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»Revenue«/mi»«mi mathvariant=¨normal¨» «/mi»«mo»§#8722;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi»Costs«/mi»«/mtd»«/mtr»«mtr»«mtd»«mi»P«/mi»«mi
mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»R«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»§#8722;«/mo»«mi»C«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi
mathvariant=¨normal¨»)«/mi»«/mtd»«/mtr»«/mtable»«/math»
A tour company offers ski excursions at «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»$«/mo»«mn»160«/mn»«/math» per person for groups up to «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»50«/mn»«/math». For groups larger than
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»50«/mn»«/math», the company will reduce the excursion cost to everyone by «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»$«/mo»«mn»2«/mn»«/math» for each person in excess of «math
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»50«/mn»«/math».
a.
What size of group would produce the maximum revenue?
b.
What ticket price would produce the maximum revenue?
c.
What is the maximum revenue?
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«/math» be the number of people in excess of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»50«/mn»«/math» where «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»§#8805;«/mo»«mn»0«/mn»«/math».
The maximum revenue is found when «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»d«/mi»«mi»R«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mn»0«/mn»«/math».
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«msup»«mi»d«/mi»«mn»2«/mn»«/msup»«mi»R«/mi»«/mrow»«mrow»«mi»d«/mi»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«mo»=«/mo»«mo»§#8722;«/mo»«mn»4«/mn»«/math» for all «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«/math», therefore a maximum revenue is produced at «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»=«/mo»«mn»15«/mn»«/math».
The company would earn a maximum revenue with a group of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»65«/mn»«/math».
The ticket cost should be «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»$«/mo»«mn»130«/mn»«/math» per person to produce the maximum revenue.
The maximum revenue is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»$«/mo»«mn»8450«/mn»«/math».
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»50«/mn»«mo»+«/mo»«mi»x«/mi»«/math» be the number of people on the trip.
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»160«/mn»«mo»-«/mo»«mn»2«/mn»«mi»x«/mi»«/math» be the reduced ticket price.
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»R«/mi»«/math» be the revenue to be maximized.
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»160«/mn»«mo»-«/mo»«mn»2«/mn»«mi»x«/mi»«/math» be the reduced ticket price.
Let «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»R«/mi»«/math» be the revenue to be maximized.
| Recall that total revenue depends on the selling price of the excursion and the number of people involved.
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»Revenue«/mi»«mo»=«/mo»«mi»cost«/mi»«mo»/«/mo»«mi»person«/mi»«mo»§#8729;«/mo»«mi»number«/mi»«mo»§#160;«/mo»«mi»of«/mi»«mo»§#160;«/mo»«mi»people«/mi»«/math»
|
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»Revenue«/mi»«mo»=«/mo»«mi mathvariant=¨normal¨»(«/mi»«mn»160«/mn»«mo»§#8722;«/mo»«mn»2«/mn»«mi»x«/mi»«mi»)(«/mi»«mn»50«/mn»«mo»+«/mo»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/math»
The maximum revenue is found when «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»d«/mi»«mi»R«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mn»0«/mn»«/math».
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mfrac»«mrow»«mi»d«/mi»«mi»R«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfenced»«mrow»«mn»160«/mn»«mo»§#8722;«/mo»«mn»2«/mn»«mi»x«/mi»«/mrow»«/mfenced»«mo»§#8729;«/mo»«mn»1«/mn»«mo»+«/mo»«mfenced»«mrow»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfenced»«mo»§#8729;«/mo»«mfenced»«mrow»«mn»50«/mn»«mo»+«/mo»«mi»x«/mi»«/mrow»«/mfenced»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»160«/mn»«mo»§#8722;«/mo»«mn»2«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»100«/mn»«mo»§#8722;«/mo»«mn»2«/mn»«mi»x«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»60«/mn»«mo»§#8722;«/mo»«mn»4«/mn»«mi»x«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd/»«mtd/»«/mtr»«mtr»«mtd»«mn»0«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»60«/mn»«mo»§#8722;«/mo»«mn»4«/mn»«mi»x«/mi»«/mtd»«/mtr»«mtr»«mtd»«mn»4«/mn»«mi»x«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»60«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»x«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»15«/mn»«/mtd»«/mtr»«/mtable»«/math»
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«msup»«mi»d«/mi»«mn»2«/mn»«/msup»«mi»R«/mi»«/mrow»«mrow»«mi»d«/mi»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«mo»=«/mo»«mo»§#8722;«/mo»«mn»4«/mn»«/math» for all «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«/math», therefore a maximum revenue is produced at «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«mo»=«/mo»«mn»15«/mn»«/math».
a.
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»number«/mi»«mi mathvariant=¨normal¨» «/mi»«mi»of«/mi»«mi mathvariant=¨normal¨» «/mi»«mi»people«/mi»«mi mathvariant=¨normal¨» «/mi»«mi»in«/mi»«mi
mathvariant=¨normal¨» «/mi»«mi»the«/mi»«mi mathvariant=¨normal¨» «/mi»«mi»group«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»50«/mn»«mo»+«/mo»«mi»x«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»50«/mn»«mo»+«/mo»«mn»15«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»65«/mn»«/mtd»«/mtr»«/mtable»«/math»
The company would earn a maximum revenue with a group of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»65«/mn»«/math».
b.
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»ticket«/mi»«mi mathvariant=¨normal¨» «/mi»«mi»cost«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»160«/mn»«mo»§#8722;«/mo»«mn»2«/mn»«mi»x«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»160«/mn»«mo»§#8722;«/mo»«mn»2«/mn»«mfenced»«mn»15«/mn»«/mfenced»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»160«/mn»«mo»§#8722;«/mo»«mn»30«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»130«/mn»«/mtd»«/mtr»«/mtable»«/math»
The ticket cost should be «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»$«/mo»«mn»130«/mn»«/math» per person to produce the maximum revenue.
c.
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»maximum«/mi»«mi mathvariant=¨normal¨» «/mi»«mi»revenue«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi mathvariant=¨normal¨»$«/mi»«mn»130«/mn»«mo»§#215;«/mo»«mn»65«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi mathvariant=¨normal¨»$«/mi»«mn»8«/mn»«mo»§#160;«/mo»«mn»450«/mn»«/mtd»«/mtr»«/mtable»«/math»
The maximum revenue is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»$«/mo»«mn»8450«/mn»«/math».
Watch the video More Maximum and Minimum Problems – Part 2 for an additional example that involves profit, revenue, and cost.
Skill Builder
For more information on revenue word problems, click the Skill Builder button to access the Skill Builder page.