6. Solve the following linear programming problem. The students in the Future Homemakers Club are making casual bags and formal bags for a fundraising project. They will line both types of bags with canvas and use leather for the handles for both bags. For the casual bags, they need 4 yards of canvas and 1 yard of leather. For the formal bags, they need 3 yards of leather and 2 yards of canvas. Their faculty advisor has purchased 56 yards of leather and 104 yards of canvas so they may use no more than what they have. The club will make a profit of $20 for each casual bag and $35 for each formal bag. Determine the number of each type of bag they need to make in order to maximize profits. a. Define your variables. b. Write the objective equation that represents the profit (P). c. Write a system of all constraints for this problem. C.. Solve the problem using the graph below. Be sure to locate all vertices and show your tests.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
icon
Related questions
icon
Concept explainers
Topic Video
Question

How do I solve this? 

# Linear Programming Problem

The following is an exercise in linear programming, a valuable approach for optimizing resources in various fields. Here, we address a problem faced by the Future Homemakers Club.

## Problem Statement

The students are tasked with creating two types of bags: casual bags and formal bags, for a fundraising project. Each bag type uses a combination of canvas and leather:

- **Casual Bags**: Require 4 yards of canvas and 1 yard of leather each.
- **Formal Bags**: Require 3 yards of leather and 2 yards of canvas each.

Their faculty advisor has provided them with 56 yards of leather and 104 yards of canvas. The profit for each bag is as follows:

- **Casual Bag**: $20
- **Formal Bag**: $35

The objective is to determine the optimal number of each type of bag to maximize profits, given the material constraints.

### Tasks

a. **Define Your Variables**

Define variables to represent the number of casual and formal bags to be produced.

b. **Write the Objective Equation**

Construct an equation to calculate the total profit (P) based on the number of each bag type produced.

c. **Write a System of Constraints**

Create inequality equations to model the limitations based on available resources (canvas and leather).

d. **Solve Using the Graph**

Analyze the provided graph to visualize and solve the system of inequalities. Identify and evaluate all potential solution points to determine the optimal number of each bag type.

### Graph Explanation

A graph is included with a coordinate grid on which the constraints can be plotted. Each axis typically represents one bag type. The feasible region, formed by the intersection of these constraints, indicates all possible combinations of bag productions that are within the material restrictions.

By evaluating the vertices of this region, you can determine which combination yields the maximum profit.

This exercise requires both algebraic and geometric methods to solve real-world optimization problems efficiently.
Transcribed Image Text:# Linear Programming Problem The following is an exercise in linear programming, a valuable approach for optimizing resources in various fields. Here, we address a problem faced by the Future Homemakers Club. ## Problem Statement The students are tasked with creating two types of bags: casual bags and formal bags, for a fundraising project. Each bag type uses a combination of canvas and leather: - **Casual Bags**: Require 4 yards of canvas and 1 yard of leather each. - **Formal Bags**: Require 3 yards of leather and 2 yards of canvas each. Their faculty advisor has provided them with 56 yards of leather and 104 yards of canvas. The profit for each bag is as follows: - **Casual Bag**: $20 - **Formal Bag**: $35 The objective is to determine the optimal number of each type of bag to maximize profits, given the material constraints. ### Tasks a. **Define Your Variables** Define variables to represent the number of casual and formal bags to be produced. b. **Write the Objective Equation** Construct an equation to calculate the total profit (P) based on the number of each bag type produced. c. **Write a System of Constraints** Create inequality equations to model the limitations based on available resources (canvas and leather). d. **Solve Using the Graph** Analyze the provided graph to visualize and solve the system of inequalities. Identify and evaluate all potential solution points to determine the optimal number of each bag type. ### Graph Explanation A graph is included with a coordinate grid on which the constraints can be plotted. Each axis typically represents one bag type. The feasible region, formed by the intersection of these constraints, indicates all possible combinations of bag productions that are within the material restrictions. By evaluating the vertices of this region, you can determine which combination yields the maximum profit. This exercise requires both algebraic and geometric methods to solve real-world optimization problems efficiently.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 3 steps with 1 images

Blurred answer
Knowledge Booster
Application of Algebra
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,