Questions and Answers
What is the constraint for the number of cappuccino bottles and café au lait bottles produced?
Complete the objective function: P = x + ___y.
0.50
Which graph represents the feasible region for the system with constraints: x + y ≥ 80, 3x + 2y ≤ 360, x + 2y ≤ 200?
C
What are the minimum and maximum values of the objective function F = 8x + 5y?
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Complete the constraints on the system: y ≤ ___, x + y ≥ ___, x ≤ ___
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What is the objective function for minimizing costs?
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What is the minimum value for minimizing the objective function P = 20x + 16y?
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Explain the steps for maximizing the objective function P = 3x + 4y.
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What objective function can be used to maximize profit? P = ___a + ___b.
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What is the maximum profit when producing units of product A and product B?
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What are the vertices of the feasible region with constraints?
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What is the minimum value of the objective function C = 4x + 9y?
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Which region represents the graph of the feasible region for the given constraints?
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Study Notes
Bottled Coffee Production
- Two types of bottled coffee: cappuccinos (6 oz coffee, 2 oz milk, $0.40 profit) and cafés au lait (4 oz coffee, 4 oz milk, $0.50 profit).
- Daily available resources: 720 ounces of coffee and 400 ounces of milk.
- Minimum requirement: at least 80 drinks produced daily.
- Variables defined: x = cappuccino bottles, y = café au lait bottles.
Objective Function
- Profit function defined as P = 0.40x + 0.50y.
Constraints for Coffee Production
- Constraints include:
- x + y ≥ 80
- 3x + 2y ≤ 360
- x + 2y ≤ 200
- x ≥ 0
- y ≥ 0.
Feasible Region
- Graph C represents the feasible region for the coffee production constraints.
- Vertices of the feasible region: (0, 100), (0, 80), (80, 60), (80, 0), (120, 0).
Objective Function Analysis
- Objective function F = 8x + 5y evaluated for minimum (400) and maximum (960).
Printing Company Constraints
- Supplier X charges $25/case, Supplier Y charges $20/case.
- Constraints include:
- At least 45 cases ordered daily.
- Maximum of 30 cases from Supplier X.
- Supplier Y can be no more than twice Supplier X’s cases.
- Variables defined: x = cases from Supplier X, y = cases from Supplier Y.
Objective Function for Printing
- Objective function to minimize costs: D = 25x + 20y.
Minimum Values
- For the objective function P = 20x + 16y, the minimum value is 780, occurring at x = 15, y = 30.
Steps for Maximization
- To maximize the objective function P = 3x + 4y:
- Graph the inequalities from constraints.
- Identify intersection points that form a polygon.
- Calculate the objective function at each point to find the maximum value.
Product A and B Production
- Product A requires at least 30 units; Product B needs at least 10 units.
- Max production limit of 80 units.
- Profit: $15 from Product A, $8 from Product B.
- Objective function for profit maximization: P = 15a + 8b.
Maximum Profit Calculation
- Feasible region vertices for products are: (70, 10), (30, 10), and (30, 50).
- Maximum profit calculated at 1130.
Feasible Region Vertices
- Provided vertices of the feasible region include (0,15), (10,15), and (20,5).
Minimum Objective Function Value
- Minimum value for the objective function C = 4x + 9y is calculated as 125.
Feasible Region Graphing
- Constraints presented: y ≥ 2x, x + y ≤ 14, y ≥ 1, 5x + y ≥ 14, x + y ≥ 9.
- Region A represents the graph of the feasible region.
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Description
Explore the optimization of bottled coffee production through an analysis of the profit function and constraints. This quiz covers key elements like resource allocation for cappuccinos and cafés au lait to maximize profits while meeting daily production requirements. Test your understanding of linear programming concepts and feasible regions.
