Math Problem Statement
Donna De Paul is raising money for the homeless. She discovers that each church group requires 2 hours of letter writing and 1 hour of follow-up, while for each labor union she needs 2 hours of letter writing and 3 hours of follow-up. Donna can raise $150 from each church group and $200 from each union local, and she has a maximum of 16 hours of letter writing and a maximum of 14 hours of follow-up available per month. Determine the most profitable mixture of groups she should contact and the most money she can raise in a month. Question content area bottom Part 1 Let x 1 be the number of church groups, and let x 2 be the number of labor unions. What is the objective function? zequals 150x 1plus 200x 2 (Do not include the $ symbol in your answers.) Part 2 She should contact enter your response here church group(s) and 2 labor union(s), to obtain a maximum of $ 14 in donations. (Simplify your answers.)
Solution
Let's break this problem into manageable parts and find the solution step by step.
Part 1: Formulating the Objective Function
Let:
- be the number of church groups Donna contacts.
- be the number of labor unions she contacts.
The objective is to maximize the amount of money Donna can raise. The amount raised is given by:
- $150 per church group:
- $200 per labor union:
Thus, the objective function (denoted as ) is:
Part 2: Constraints
Next, we establish the constraints based on the time limits for letter writing and follow-up:
- Letter writing constraint: Each church group requires 2 hours, and each labor union requires 2 hours of letter writing. Donna has a maximum of 16 hours available for letter writing. This gives us the constraint:
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Math Problem Analysis
Mathematical Concepts
Linear Programming
Optimization
Inequalities
Algebra
Formulas
Objective function: z = 150x1 + 200x2
Constraint 1 (Letter writing): 2x1 + 2x2 ≤ 16
Constraint 2 (Follow-up): 1x1 + 3x2 ≤ 14
Theorems
Simplex Method (for optimization)
Feasibility and Boundedness in Linear Programming
Suitable Grade Level
Grades 10-12
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