Math Problem Statement

A company has a budget constraint for two projects: Project A and Project B. The relationship between the budgets allocated to these projects is represented by the following inequalities: 2x + 3y ≤ 60 x + 2y ≤ 30 (Where x is the budget for Project A in thousands and y is the budget for Project B in thousands) 1. Graph the inequalities on the Cartesian plane. 2. Identify the feasible region. 3. What are the maximum budget allocations for Projects A and B that meet these constraints?

Solution

The graph above represents the feasible region for the budget constraints of Projects A and B. The feasible region, shaded in gray, is bounded by the lines for the inequalities 2x+3y602x + 3y \leq 60 and x+2y30x + 2y \leq 30.

Feasible Region Vertices

The vertices of the feasible region, where the maximum budget allocations are found, are:

  1. (0,15)(0, 15)
  2. (30,0)(30, 0)
  3. (12,9)(12, 9)

These points represent potential allocations for Projects A and B that meet the constraints.

Maximum Budget Allocations

To determine the maximum budget allocation, we can evaluate these vertices based on any additional optimization criteria, such as maximizing a specific objective function (e.g., maximizing x+yx + y if both budgets are to be maximized jointly). In this case, without a specific objective function, these points serve as the boundary conditions within which any allocation would meet the constraints.

If you need further assistance with optimization or have additional questions, please let me know.


Would you like more details or have any specific questions on this solution? Here are some related questions for further exploration:

  1. How do we determine if a point lies within the feasible region without graphing?
  2. What if we wanted to minimize the combined budget for Projects A and B?
  3. How would the feasible region change if the inequalities were strict (i.e., << instead of \leq)?
  4. Can additional constraints be added to further refine the feasible region?
  5. How would we interpret these inequalities if the context changed, such as in production rather than budgeting?

Tip: When working with linear inequalities, always identify intercepts first for easier graphing and visualization of feasible regions.

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Math Problem Analysis

Mathematical Concepts

Linear Inequalities
Graphing Inequalities
Feasible Region

Formulas

2x + 3y ≤ 60
x + 2y ≤ 30

Theorems

Linear programming feasibility and optimization

Suitable Grade Level

Grades 10-12