cKendra Kilmer February 5, 2010Section 3.1 - Graphing Systems of Linear InequalitiesExample 1: Find the graphical solution of the inequality y − x ≤ 0.Example 2: Find the graphical solution of the inequality 5x − 3y < 15.1cKendra Kilmer February 5, 2010Example 3: Determine graphically the solution set for the system of inequalities:2x + 4y > 16−x + 3y ≥ 72cKendra Kilmer February 5, 2010Example 4: Determine graphically the solution set for the system of inequalities:x + y ≤ 42x + y ≤ 62x − y ≥ −1x ≥ 0, y ≥ 0Definition: A solution set of a system of linear inequalities is bounded if it can be enclosed by a circle.Otherwise, it is unbounded.3cKendra Kilmer February 5, 2010Section 3.2 - Linear Programming ProblemsDefinition: A linear programming problem consists of a linear objective function to be maximized orminimized subject to certain constraints in the form of linear equalities or inequalities.Example 1: A farmer plans to plant two crops, A and B. The cost of cultivating crop A is $50/acre,whereas that of crop B is $20/acre. The farmer has a maximum of $4,000 available for land cultivation.Each acre of crop A requires 75 labor-hours, and each acre of crop B requires 150 labor-hours. Thefarmer has a maximum of 15,000 labor-hours available. If she expects to make a profit of $90/acre oncrop A and $100/acre on crop B, how many acres of each crop should she plant in order to maximizeher profit? (Set-up the linear programming problem.)Example 2: Deluxe River Cruises operates a fleet of river vessels. The fleet has two types of vessels:A type-A vessel has 60 deluxe cabins and 96 standard cabins, whereas a type-B vessel has 40 deluxecabins and 160 standard cabins. Under a charter agreement with Odyssey Travel Agency, Deluxe RiverCruises is to provide Odyssey with a minimum of 240 deluxe and 672 standard cabins for their 15-daycruise in May. It costs $44, 000 to operate a type-A vessel and $54, 000 to operate a type-B vessel forthat period. How many of each type vessel should be used in order the keep the operating costs to aminimum? (Set-up the linear programming problem.)4cKendra Kilmer February 5, 2010Example 3: The manager of food sales for a certain music festival is trying to determine how muchfood they should buy/sell. They have decided to charge $5 for each hot dog, $6 for each hamburger,and $4 for each bucket of popcorn. They have determined that it will cost them $0.50 to purchase eachhot dog, $1.00 to purchase each hamburger, and $0.25 to purchase each bucket of popcorn. They have$5,000 available to purchase the food, have determined that they should purchase at least three timesas many hotdogs as hamburgers and buckets of popcorn combined, and they have determined that theyshould purchase no more than twice as many buckets of popcorn as hamburgers. Assuming that theysell all of the food they purchase, how many hotdogs, hamburgers, and buckets of popcorn should theypurchase in order to maximize their revenue? (Set-up the linear programming problem.)5cKendra Kilmer February 5, 2010Section 3.3 - Method of CornersExample 1: Let’s look at solving Example 1 from Section 3.2. We found that the linear programmingproblem was:Maximize P = 90x + 100ySubject to 50x + 20y ≤ 4,00075x + 150y ≤ 15,000x ≥ 0, y ≥ 06cKendra Kilmer February 5, 2010Method of Corners:1. Graph the corresponding equations by hand or on your calculator (i.e. solve each inequality for yand replace the inequality sign with an equal sign). Be careful in choosing your window. If thereare non-negativity constraints on x and y, account for these by choosing x-min and y-min to be 0.2. Determine the feasible region.3. Find the coordinates of the corner points. You can either do this algebraically or on the calculator.There are four possible corner points:– The origin (i.e the point (0, 0))– The intersection of two lines (use 2nd TRACE 5:intersect)– The intersection of a line and the y-axis (recognize that this is the y-intercept of the line or use2nd TRACE 1:value and enter in 0 for x)– The intersection of a line and the x-axis (plug in 0 for y and solve for x or use 2nd TRACE2:zero and specify a left and right bound that includes the point where the line actually crossesthe x-axis)4. Make a table and evaluate the objective function at each corner point.Example 2: Let’s solve Example 2 from Section 3.2. We found that the linear programming problemwas:Minimize C = 44,000x + 54,000ySubject to 60x + 40y ≥ 24096x + 160y ≥ 672x ≥ 0, y ≥ 07cKendra Kilmer February 5, 2010Example 3: Solve the following linear programming problemMinimize P = 2x + 2ySubject to 2x + 3y ≤ 30y − x ≤ 5x + y ≥ 5x ≤ 10x ≥ 0, y ≥ 08cKendra Kilmer February 5, 2010Example 4: You manage an ice cream factory that makes two flavors: Creamy Vanilla and ContinentalMocha. Each quart of Creamy Vanilla requires 2 eggs and 3 cups of cream. Each quart of ContinentalMocha requires 1 egg and 3 cups of cream. You have in stock 500 eggs and 900 cups of cream. Youmake a profit of $3 on each quart of Creamy Vanilla and $4 on each quart of Continental Mocha.a) How many quarts of each flavor should you make in order to earn the largest profit?b) Are there any leftover resources? Be
View Full Document
Unlocking...