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:

  • x1x_1 be the number of church groups Donna contacts.
  • x2x_2 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: 150x1150x_1
  • $200 per labor union: 200x2200x_2

Thus, the objective function (denoted as zz) is: z=150x1+200x2z = 150x_1 + 200x_2

Part 2: Constraints

Next, we establish the constraints based on the time limits for letter writing and follow-up:

  1. 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: 2x1+2x2162x_1 + 2x_2 \leq 16

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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