10.8 Day 2 Linear Programming 2011
February 25, 2011
10.8 Linear Programming
DAY 2
Warm­up:
Solve each system of inequalities by graphing.
1. x > 5
y > ­3x + 6
2. 3y > 5x + 2 3. x + 3y < ­6
y < ­x + 7 2x ­ 3y < 4
Objectives: • Set up linear programming problems.
• Solve linear programming problems.
Aug 14­9:04 PM
Aug 14­9:31 PM
Example #2: 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 ordered
Let y = number of cases of roasted peanuts ordered
Let P = total profit
Write:
Use the given information to write the objective function and the constraints on the problem.
Objective Function: Mixed Nuts
Roasted Peanuts
12 cans per case
20 packages per case
You pay...$24 per case
Sell at...$3.50 per can
You pay...$15 per case
Sell at...$1.50 per package
$18 profit per case!
Example #2: 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?
P = 18x + 15y
$15 profit per case!
Constraints:
Mixed Nuts
x > 0
y > 0
12x + 20y < 500
24x + 15y < 600
Aug 14­9:44 PM
Roasted Peanuts
12 cans per case
20 packages per case
You pay...$24 per case
Sell at...$3.50 per can
You pay...$15 per case
Sell at...$1.50 per package
$18 profit per case!
$15 profit per case!
Sep 28­4:50 PM
Example #2: 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?
Example #2: 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?
Now follow the steps from the first example.
Step 3: Evaluate the profit P at each vertex. Step 1: Graph the constraints.
x > 0
y > 0
12x + 20y < 500
24x + 15y < 600
P = 18x + 15y
50
(0, 0)
(25, 0)
(15, 16)
(0, 25)
40
30
Step 2: Find the vertices.
(0, 0)
(25, 0)
(15, 16)
(0, 25)
P = 18(0) + 15(0) = 0
P = 18(25) + 15(0) = 450
P = 18(15) + 15(16) = 510
P = 18(0) + 15(25) = 375
20
Step 4: State the results in complete sentences.
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You can maximize your profit by selling 15 cases of mixed nuts and 16 cases of roasted peanuts. The maximum profit is $510.
Sep 28­4:50 PM
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Sep 28­4:50 PM
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10.8 Day 2 Linear Programming 2011
February 25, 2011
Homework: page 820 Sep 28­5:38 PM
Feb 18­12:38 PM
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