3.7 Optimization Problems

Transcription

1 37 Optimization Problems Applied Maximum and Minimum Problems minimize cost maximize profit minimize waste least time Def optimization means finding where some function (model) has its greatest or smallest value 1

2 Concept: (Section 31) If f is continuous on the closed interval, then where f has a maximum or minimum must be one of the following: 1 interior points where f ' = 0, 2 interior points where f ' does NOT exist, 3 endpoints of the function's domain Note: none of these points is necessarily the location of an extreme value, but these are the only candidates! a b a b a b critical numbers endpoints 2

3 Defs primary equation an equation/formula for the quantity to be optimized secondary equation an equation/formula relating the independent variables of the primary equation Purpose: to reduce the primary equation to one having a single independent variable before attempting to maximize or minimize it feasible domain the values of "x" that make sense in the problem 3

4 Guidelines for Solving Applied Minimum and Maximum Problems (p 212) Identify all given quantities and quantities to be determined When feasible, make a sketch Write a primary equation for the quantity that is to be maximized (or minimized) (A review of several useful formulas from geometry is presented inside the front or back cover) Reduce the primary equation to one having a single independent variable This may involve the use of secondary equations relating the independent variables of the primary equation Determine the feasible domain of the primary equation That is, determine the values for which the stated problem makes sense Determine the desired maximum or minimum value by the calculus techniques discussed in Sections 31 through 34 4

5 examples: 1 Find two numbers whose sum is 100 and whose product is a maximum 5

6 2 Product Design: An open top box is to be made by cutting small congruent squares from the corners of a 20 by 25 inch sheet of tin and bending up the sides Use analytic calculus to determine how large the squares cut from the corners should be to make the box hold as much as possible, the resulting maximum value, and support your answer graphically 20" 25" x x x x x x x x 6

7 and support your answer graphically 2 7

8 3 Minimum Distance: [p216 #12] f (x) = Find the point on the graph of that is closest to the point (2, 0) x 8 (2, 0) 8

9 4 Area: [p218 #30] A rectangular page is to contain 36 square inches of print The margins on each side are to be 1 1 / 2 inches Find the dimensions of the page such that the least amount of paper is used 1 1 /2 " 1 1 /2 " "You start with a dream But without hard work, you may end up with nothing but the dream Hold onto your dreams 1 1 " They're very /2 1 1 " /2 important The ideal situation, however, is to go out and make those dreams come true 1 1 /2 " "You start with a dream But without hard work, 1 1 " you may /2 1 1 " end up with /2 nothing but the dream Hold onto your dreams They're very important The ideal situation, however, is to go out and make those 1 1 " /2 1 1 " /2 "You start with a dream But without hard work, you may end up with nothing but the dream Hold onto your dreams 1 1 " They're very important The ideal " /2 situation, however, is to go out and make 11 /2 those dreams come true 1 1 " /2 1 1 / 2 " 1 1 / 2 " 1 1 /2 " 1 1 / 2 " 9

10 5 Geometry: A rectangular is to be inscribed in a semicircle of radius 2 Find the largest area of the rectangle and its dimensions analytically Support your answer with a graphing utility ( 2, 0) (0, 2) (2, 0) ( 2, 0) (0, 2) (x, y) (2, 0) (0, 2) ( 2, 0) (x, y) (2, 0) 10

11 and support your answer graphically 5 11

12 6 Minimum Time: Agent 00 is waiting at the edge of a straight canal 1 mile wide, in a motorboat capable of going 40 mph There is a straight road along the opposite side of the canal where her partner will have a 50 mph motorcycle waiting for her wherever she lands At midnight, she will receive a package to be delivered to a man in a Mercedes Benz 5 miles down the road Mission: Problem: Deliver the package in the shortest possible time Where should her partner park the motorcycle? 00 1 mile 00 Ȧ MB 5 miles C 12

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14 7 Minimum Cost: A powerhouse is on one edge of a straight river and a factory is on the other edge, 100 meters downstream The river is 50 meters wide It costs $10 per meter to run electrical cable across the river and $7 per meter on land How should the cable be installed to minimize the cost? 100 m F 50 m ṖH 14

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16 Assignment: example 1 p #1, 3, 5, 7, 9, example 2 #17 20, 23 example 3 #11, 13, 29, 27, #31, 33, 39, 43 p #22, 44, 46, 49 #50(a & b), 54 example 4 example 5 examples 6 and 7 16