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Problem 8 A box is required to have a volume of 10,000 cubic inches. Material for the bottom costs 25 cents/in?, material for the sides cost 10 cents/in?, and the material for the top is 5 cents/in?. How do you minimize cost?

Problem 8 A box is required to have a volume of 10,000 cubic inches. Material for the bottom costs 25 cents/in?, material for the sides cost 10 cents/in?, and the material for the top is 5 cents/in?. How do you minimize cost?

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Problem 8** A box is required to have a volume of 10,000 cubic inches. Material for the bottom costs 25 cents/in², material for the sides cost 10 cents/in², and the material for the top is 5 cents/in². How do you minimize cost? --- This problem involves optimizing the cost of materials for constructing a box with a specific volume constraint. **Key Points for Solution:** 1. **Understand the Volume Constraint**: - The box must have a volume of 10,000 cubic inches. 2. **Identify Cost Factors**: - Bottom Material: 25 cents per square inch. - Side Material: 10 cents per square inch. - Top Material: 5 cents per square inch. 3. **Objective**: - Minimize the total cost while ensuring the volume requirement is met. 4. **Approach**: - Set up equations for the volume and cost. - Use calculus or algebraic methods to find dimensions that minimize the total cost. This problem is suitable for illustrating concepts in optimization, especially using calculus to find minimum values given certain constraints.
Transcribed Image Text:**Problem 8** A box is required to have a volume of 10,000 cubic inches. Material for the bottom costs 25 cents/in², material for the sides cost 10 cents/in², and the material for the top is 5 cents/in². How do you minimize cost? --- This problem involves optimizing the cost of materials for constructing a box with a specific volume constraint. **Key Points for Solution:** 1. **Understand the Volume Constraint**: - The box must have a volume of 10,000 cubic inches. 2. **Identify Cost Factors**: - Bottom Material: 25 cents per square inch. - Side Material: 10 cents per square inch. - Top Material: 5 cents per square inch. 3. **Objective**: - Minimize the total cost while ensuring the volume requirement is met. 4. **Approach**: - Set up equations for the volume and cost. - Use calculus or algebraic methods to find dimensions that minimize the total cost. This problem is suitable for illustrating concepts in optimization, especially using calculus to find minimum values given certain constraints.
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