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A factory that makes statuettes is trying to maximize its profit. Their two primary products are the Venus and Colossus statuettes. Each Venus requires 40 hours of machine time to rough cut the shape and 60 hours of craftsman time to finish the details. The Colossus requires 80 hours on the machines and 30 hours with the craftsman's hand. The factory is limited each day to 800 hours of machine time and 600 hours of craftsman time. Both use 12 pounds of stone, and the factory has 144 pounds of stone available per day. If each Venus brings in a profit of seventy-one dollars, and each Colossus brings in ninety-seven dollars, how many statuettes of each type should the factory make each day? Constraints: 40V+80C 800 60V+30C 600 12V+12C 144 Objective: Profit-71V +970 Graph: Solution: Venus: Colossus 80 14 15 4

A factory that makes statuettes is trying to maximize its profit. Their two primary products are the Venus and Colossus statuettes. Each Venus requires 40 hours of machine time to rough cut the shape and 60 hours of craftsman time to finish the details. The Colossus requires 80 hours on the machines and 30 hours with the craftsman's hand. The factory is limited each day to 800 hours of machine time and 600 hours of craftsman time. Both use 12 pounds of stone, and the factory has 144 pounds of stone available per day. If each Venus brings in a profit of seventy-one dollars, and each Colossus brings in ninety-seven dollars, how many statuettes of each type should the factory make each day? Constraints: 40V+80C 800 60V+30C 600 12V+12C 144 Objective: Profit-71V +970 Graph: Solution: Venus: Colossus 80 14 15 4

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Maximizing Profit for Statue Production: A Case Study** A factory that makes statuettes is trying to maximize its profit. Their two primary products are the Venus and Colossus statuettes. Each Venus requires 40 hours of machine time to rough out the shape and 10 hours of craftsman's time to finish the details. The Colossus requires 80 hours on the machines and 30 hours with the craftsman’s hand. The factory is limited each day to 600 hours of machine time and 600 hours of craftsman time. In addition, it uses 12 pounds of stone, and the factory has 144 pounds of stone available per day. If each Venus brings in a profit of seventy-one dollars, and each Colossus brings in ninety-seven dollars, how many statuettes of each type should the factory make each day? ### Constraints: - \( 40V + 80C \leq 800 \) (Machine hours) - \( 60V + 30C \leq 600 \) (Craftsman hours) - \( 12V + 12C \leq 144 \) (Pounds of stone) ### Objective: - **Profit**: \( 71V + 97C \) ### Graph Explanation: A graph is provided to show the feasible region for the constraints. The feasible region (shaded in blue) represents all the possible combinations of Venus (V) and Colossus (C) that the factory can produce without exceeding the given resources of machine hours, craftsman hours, and pounds of stone. The x-axis represents the number of Venus statuettes (V), and the y-axis represents the number of Colossus statuettes (C). Each line on the graph represents one of the constraints: - The line closer to the horizontal axis corresponds to the machine time constraint: \(40V + 80C = 800\). - The line sloping down more steeply corresponds to the craftsman time constraint: \(60V + 30C = 600\). - The line with the gentlest slope corresponds to the stone constraint: \(12V + 12C = 144\). The area where these three constraints overlap is the feasible region. The vertices of the feasible region should be evaluated to identify the maximum profit configuration. ### Solution: - **Number of Venus statuettes (V)**: - **Number of Colossus statuettes (
Transcribed Image Text:**Maximizing Profit for Statue Production: A Case Study** A factory that makes statuettes is trying to maximize its profit. Their two primary products are the Venus and Colossus statuettes. Each Venus requires 40 hours of machine time to rough out the shape and 10 hours of craftsman's time to finish the details. The Colossus requires 80 hours on the machines and 30 hours with the craftsman’s hand. The factory is limited each day to 600 hours of machine time and 600 hours of craftsman time. In addition, it uses 12 pounds of stone, and the factory has 144 pounds of stone available per day. If each Venus brings in a profit of seventy-one dollars, and each Colossus brings in ninety-seven dollars, how many statuettes of each type should the factory make each day? ### Constraints: - \( 40V + 80C \leq 800 \) (Machine hours) - \( 60V + 30C \leq 600 \) (Craftsman hours) - \( 12V + 12C \leq 144 \) (Pounds of stone) ### Objective: - **Profit**: \( 71V + 97C \) ### Graph Explanation: A graph is provided to show the feasible region for the constraints. The feasible region (shaded in blue) represents all the possible combinations of Venus (V) and Colossus (C) that the factory can produce without exceeding the given resources of machine hours, craftsman hours, and pounds of stone. The x-axis represents the number of Venus statuettes (V), and the y-axis represents the number of Colossus statuettes (C). Each line on the graph represents one of the constraints: - The line closer to the horizontal axis corresponds to the machine time constraint: \(40V + 80C = 800\). - The line sloping down more steeply corresponds to the craftsman time constraint: \(60V + 30C = 600\). - The line with the gentlest slope corresponds to the stone constraint: \(12V + 12C = 144\). The area where these three constraints overlap is the feasible region. The vertices of the feasible region should be evaluated to identify the maximum profit configuration. ### Solution: - **Number of Venus statuettes (V)**: - **Number of Colossus statuettes (
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