Chapter Linear Programs     Section 3.1 Linear Inequalities in Two Variables Section 3.2 Solutions of Systems of Inequalities: A Geometric Picture Section 3.3 Linear Programming: A Geometric Ap proach Section 3.4 Applications Graph the linear inequality Graph the linear inequality Graph the linear inequality Graph the linear inequality Section 3.1 Graphing a Linear Inequality Graph the inequality of the form ax + by < c (The procedure also applies if the inequality symbols are or >.) Select a point that is not on the line from one half plane The point (0,0) is usually a good choice when it is not on the line If (0,0) is on the line If (0,0) is on the line, use a point that is not on the line Continued on next slide Continued Substitute the coordinates of the point for x and y in the inequality a) If the selected point satisfies the inequality, then shade the half plane where the point lies These points are on the graph b) If the selected point does not satisfy the inequality, shade the half plane opposite the point c) If the inequality symbol is < or >, use a dotted line for the graph of ax + by = c This indicates that the points on the line are not a part of the graph d) If the inequality symbol is < or >, use a solid line for the graph of ax + by = c This indicates that the line is a part of the graph Example An automobile assembly plant has an assembly line that produces the Hatchback Special and the Sportster Each Hatchback requires 2.5 hours of assembly line time, and each Sportster requires 3.5 hours The assembly line has a maximum operating time of 140 hours per week Graph the number of cars of each type that can be produced in one week A bakery is making whole-wheat bread and apple bran muffins The bread takes hours to prepare The muffins take 0.5 hour to prepare The maximum preparation time available is 16 hours Graph the number of of each type that can be prepared in one day Discussion and Writing  Explain in your own words what a linear programming problem is and how it can be solved HW 3.3 Pg 231-239 1-75 odd Section 3.4 Applications     Solving linear programming problems geometrically works well when there are only two variables and a few constraints Typically though, linear programming problems will require dozens of variables with several constraints A correct analysis and description of the problem is essential before applying any method An erroneous constraint will yield an erroneous solution This section works on correctly setting up linear programming problems in more than two variables so that the methods in Chapter can be utilized to solve these larger systems Example Adventure Time offers one-week summer vacations during the month of August The package includes round-trip transportation and a week’s accommodations at the Lodge The Lodge gives a discount to Adventure Number of Condos Time if they rent two- or three-week blocks Week needed of condos, rent of $1000 per condo for a 30 two-week period and $1300 per condo for a First 3-week period Adventure Time expects Second 42 to need the number of condos shown Third 21 Fourth 32 How many condos should Adventure Time rent for two weeks, and how many should be rented for three weeks, to meet the needed number and to minimize Adventure Time’s rental costs? Example continued SOLUTION First, determine all possible ways to schedule two- and threeweek blocks in August The table below helps to “visualize” the possible ways to schedule the blocks Two-week periods Week Week Week Week Three-week periods Number of condos needed Example continued Since Adventure Time wants to minimize their rental costs, we need to minimize The relationship between the number of condos needed and the five time periods can be written as HW 3.4 Pg 243-246 1-20 ... plant has an assembly line that produces the Hatchback Special and the Sportster Each Hatchback requires 2.5 hours of assembly line time, and each Sportster requires 3.5 hours The assembly line... kind of bicycle The table gives the hours of general labor, machine time, and technical labor required to make one bicycle in each plant For the two plants combined, the manufacturer can afford... manufacturer can afford to use up to 4000 hours of general labor, up to 1500 hours of machine time, and up to 2300 hours of technical labor per week Write some linear inequalities that describe this situation