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3.4 Day 2 Linear Programming 2010
1
October 27, 2010
Aug 149:04 PM
3.4 Linear Programming
DAY 2
Objectives: • To find maximum and minimum values• To solve problems with linear programming
3.4 Day 2 Linear Programming 2010
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October 27, 2010
Aug 149:31 PM
Check Skills You'll Need:
Solve each system of inequalities by graphing.
1. x > 5 2. 3y > 5x + 2 3. x + 3y < 6 y > 3x + 6 y < x + 7 2x 3y < 4
3.4 Day 2 Linear Programming 2010
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October 27, 2010
Aug 149:44 PM
Solving RealWorld Problems:Suppose you are selling cases of mixed nuts and roasted peanuts. You can order no more than a total of 500 cans and packages and spend no more than $600. How can you maximize your profit? How much is the maximum profit?
Mixed Nuts Roasted Peanuts12 cans per case 20 packages per case
You pay...$24 per case You pay...$15 per caseSell at...$3.50 per can Sell at...$1.50 per package
$18 profit per case! $15 profit per case!
3.4 Day 2 Linear Programming 2010
4
October 27, 2010
Sep 284:50 PM
Solving RealWorld Problems:Suppose you are selling cases of mixed nuts and roasted peanuts. You can order no more than a total of 500 cans and packages and spend no more than $600. How can you maximize your profit? How much is the maximum profit?
Define: Let x = number of cases of mixed nuts orderedLet y = number of cases of roasted peanuts orderedLet P = total profit
Relate: Organize the information into a table
Mixed Nuts Roasted Peanuts TotalNumber of Cases x y x + yNumber of Units 12x 20y 500 constraintCost 24x 15y 600 constraintProfit 18x 15y 18x + 15y objective
3.4 Day 2 Linear Programming 2010
5
October 27, 2010
Sep 284:50 PM
Solving RealWorld Problems:Suppose you are selling cases of mixed nuts and roasted peanuts. You can order no more than a total of 500 cans and packages and spend no more than $600. How can you maximize your profit? How much is the maximum profit?
Mixed Nuts Roasted Peanuts TotalNumber of Cases x y x + yNumber of Units 12x 20y 500 constraintCost 24x 15y 600 constraintProfit 18x 15y 18x + 15y objective
Write: Write and simplify the constraints. Write the objective function.
12x + 20y < 500 3x + 5y < 12524x + 15y < 600 ⇒ 8x + 5y < 200 P = 18x + 15yx > 0, y > 0 x > 0, y > 0
Now follow the steps from yesterday to determine what values of x and y maximize your profit.
{ {
3.4 Day 2 Linear Programming 2010
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October 27, 2010
Sep 284:50 PM
Step 1: Graph the constraints (solve for y first)
3x + 5y < 125 y < 3/5x + 258x + 5y < 200 ⇒ y < 8/5x + 40x > 0, y > 0 x > 0, y > 0
10
10 20
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30
30 40
40
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50
{ {
Step 2: Find the coordinates of each vertex
(0, 0)(25, 0)(15, 16)(0, 25)
3.4 Day 2 Linear Programming 2010
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October 27, 2010
Sep 284:50 PM
Step 3: Evaluate P at each vertex
P = 18x + 15y
(0, 0) P = 18(0) + 15(0) = 0(25, 0) P = 18(25) + 15(0) = 450(15, 16) P = 18(15) + 15(16) = 510(0, 25) P = 18(0) + 15(25) = 375
Step 4: State the results in complete sentences.
You can maximize your profit by selling 15 cases of mixed nuts and 16 cases of roasted peanuts. The maximum profit is $510.
3.4 Day 2 Linear Programming 2010
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October 27, 2010
Sep 285:38 PM
Homework: page 142 (10, 11, 20, 21, 23 27)