Problem 4 - Finite Mathematics and Applied Calculus

Question

\(\begin{array}{ll}\text { Maximize } & p=2 x+3 y \\ \text { subject to } & 3 x+8 y \leq 24 \\ & 6 x+4 y \leq 30 \\ & x \geq 0, y \geq 0 .\end{array}\)

Step-by-Step Solution

Answer

The maximum value of the objective function \(p = 2x + 3y\) is \(22.5\), and it occurs at the point \((0,7.5)\) subject to the given constraints.

Step 1: Identify the Constraints and Objective Function

We have the following constraints: 1. \(3x + 8y \leq 24\) 2. \(6x + 4y \leq 30\) 3. \(x \geq 0\) 4. \(y \geq 0\) And the objective function to maximize: \[p = 2x + 3y\]

Step 2: Graph the Feasible Region

Graph the inequalities on the coordinate plane to find the feasible region, which is the area where all constraints are satisfied simultaneously. 1. For \(3x + 8y \leq 24\), we first find the boundary by setting the inequality as an equation: \(3x + 8y = 24\). 2. For \(6x + 4y \leq 30\), we do the same: \(6x + 4y = 30\). 3. The non-negativity constraints are given by the axes: \(x \geq 0\) and \(y \geq 0\). By graphing these equations, we find the feasible region, which is an enclosed polygon on the first quadrant.

Step 3: Identify the Corner Points

The corner points are the vertices of the feasible region. We find these points by solving the system of linear equations formed by the intersections of the boundary equations. There are four such systems: 1. \(x = 0\) and \(3x + 8y = 24\). 2. \(x = 0\) and \(6x + 4y = 30\). 3. \(y = 0\) and \(3x + 8y = 24\). 4. \(y = 0\) and \(6x + 4y = 30\). Solving these systems, we find the corner points: \((0,3), (0,7.5), (4,0)\), and \((5,0)\).

Step 4: Evaluate the Objective Function at Corner Points

We now evaluate the objective function, \(p = 2x + 3y\), at each corner point to find the maximum value: 1. \(p(0,3) = 2(0) + 3(3) = 9\) 2. \(p(0,7.5) = 2(0) + 3(7.5) = 22.5\) 3. \(p(4,0) = 2(4) + 3(0) = 8\) 4. \(p(5,0) = 2(5) + 3(0) = 10\) The maximum value of the objective function is \(22.5\) at the point \((0,7.5)\).

Step 5: Final Answer

The maximum value of the objective function \(p = 2x + 3y\) is \(22.5\), and it occurs at the point \((0,7.5)\) subject to the given constraints.

Key Concepts

Objective Function

In linear programming, the objective function is vital, serving as the mathematical description of the goal you want to achieve. In this exercise, the objective is to maximize the value given by the equation \(p = 2x + 3y\). This equation combines two variables, \(x\) and \(y\), each with their coefficients indicating their contribution to the total value.The coefficients, \(2\) for \(x\) and \(3\) for \(y\), show the rate of contribution of each variable to the objective. Your task is to find the combination of \(x\) and \(y\) within the feasible region that yields the highest possible value for \(p\). Key points:

• The objective function should always align with your goal, typically either maximizing or minimizing.
• In business, this could be about maximizing profit or minimizing costs.

Understanding how coefficients affect the objective function can greatly enhance your strategic decision-making in reaching the optimal solution.

Feasible Region

The feasible region represents all possible combinations of variables that satisfy given constraints in a linear programming model. In graphical terms, it is the area where all constraints overlap on a coordinate plane.To find the feasible region for this problem, you need to plot the constraints:

• \(3x + 8y \leq 24\)
• \(6x + 4y \leq 30\)
• \(x \geq 0\)
• \(y \geq 0\)

Once plotted, the feasible region will be that closed, bounded polygon in the first quadrant. Key points to remember:

• The feasible region must be non-empty and bounded for a solution to exist.
• Every point within the feasible region is a potential solution, but the optimal one lies at a vertex.

This region determines the limits within which the solutions can vary and can include a complex shape depending on constraints.

Corner Points

Corner points, or vertices, are the points where the edges of the feasible region intersect. In linear programming, these points are crucial because, according to the theory, if there is an optimal solution, it will always be found at one of the corner points.In this case, to find the corner points, solve the system of linear equations where the constraint lines intersect:

• \(x = 0\) and \(3x + 8y = 24\)
• \(x = 0\) and \(6x + 4y = 30\)
• \(y = 0\) and \(3x + 8y = 24\)
• \(y = 0\) and \(6x + 4y = 30\)

Solving these, we find corner points \((0,3), (0,7.5), (4,0), (5,0)\).Why corner points?

• They are easier to calculate and evaluate than every point in the region.
• Linear functions achieve their maxima and minima at the vertices for bounded regions.

Constraints

Constraints define the limitations or requirements that must be followed in the linear programming problem. They limit the choices available to the decision-maker. In our problem, the constraints are the inequalities that confine our decision variables, \(x\) and \(y\), to a specific range.For this exercise, the constraints are:

• \(3x + 8y \leq 24\)
• \(6x + 4y \leq 30\)
• \(x \geq 0\)
• \(y \geq 0\)

These linear inequalities collectively form a region on the graph where any point within satisfies all constraints simultaneously.Important points:

• Constraints can include resource limitations, budget restrictions, or non-negativity requirements.
• They must be linear in linear programming, resulting in manageable graphical representation.

Understanding constraints helps you understand what is possible in the problem context and shapes the feasible region.