Problem 23 - Precalculus with Limits

Question

Finding Minimum and Maximum Values, find the minimum and maximum values of the objective function and where they occur, subject to the constraints \(x \geq 0, y \geq 0,3 x+y \leq 15\) and \(4 x+3 y \leq 30 .\) $$ z=x+y $$

Step-by-Step Solution

Answer

The maximum value of the function is 10 at the point (0,10), and the minimum value is 0 at the point (0,0).

Step 1: Identify the Constraints

The problem provides four constraints: \(x \geq 0\), \(y \geq 0\), \(3x + y \leq 15\), and \(4x + 3y \leq 30\). The first two constraints means that x and y must be positive or zero. The next two constraints can be graphed to visualize the feasible region.

Step 2: Graph the Constraints

Prepare a coordinate system and start to graph each constraint. First, solve \(3x + y = 15\) to find where the line intersects the x and y axes, which are at (5,0) and (0,15). Plot these points and draw a line between them. The inequality \(3x + y \leq 15\) means we take points below this line. Do the same for \(4x + 3y = 30\), which intersects x-axis at (7.5,0) and y-axis at (0,10). In this case, feasible region is also below the line. The overlap of these two regions will form the feasible region.

Step 3: Find the Corners of the Feasible Region

The feasible region is a polygon formed by the intersection of all constraints. Identify its corners, as the maximum and minimum values of the objective function must occur at one of these corner points. In this case, corners are at the points (0,0), (0,10), (5,0) and intersection point of the lines \(3x + y = 15\) and \(4x + 3y = 30\). Solve these two equalities simultaneously to find this intersection point, which is (3,6).

Step 4: Evaluate Objective Function at the Corners

Substitute each corner point into the objective function \(z = x + y\). The corner values are (0,0), (0,10), (5,0), and (3,6). This results in z values of 0, 10, 5, and 9 respectively.

Step 5: Identify the Maximum and Minimum Values

The maximum value of the objective function is 10 at the point (0,10), and the minimum value is 0 at the point (0,0). These are the highest and lowest z-values we found in previous step.

Key Concepts

Objective Function

In linear programming, the objective function is a mathematical expression that defines what you want to optimize. This could mean either maximizing or minimizing a certain quantity. In the context of our exercise, the objective function is given by the formula \( z = x + y \). This represents a linear combination of variables \( x \) and \( y \).

The overarching goal when working with an objective function is to determine the maximum or minimum values that \( z \) can take. To do this, we need to consider certain conditions or restrictions, known as constraints, impacting our decision space. Once these are identified, the next step is to evaluate the function at specific points within a defined region, to find the optimal values.

• Objective functions can be any linear expression of decision variables.
• Finding the optimal value means looking for either the highest or lowest point, depending on the problem.

In our exercise, we are tasked with finding both the minimum and maximum values of \( z = x + y \) given certain constraints. This involves exploring feasible solutions and the interplay of constraints to yield the desired outcome.

Feasible Region

The feasible region in linear programming is one of the most critical areas you need to identify. It represents the set of all possible points, or solutions, that satisfy the given constraints of the problem.

In our exercise, the feasible region is bounded by the constraints \( x \geq 0 \), \( y \geq 0 \), \( 3x + y \leq 15 \), and \( 4x + 3y \leq 30 \). These constraints form a polygon on a graph, and the area inside this polygon is the feasible region.

• The feasible region is crucial as it's the only place where the optimal solution for the objective function can be found.
• It provides a visual representation to better understand the intersecting limitations.

By analyzing this region, we can easily visualize possible solutions that adhere to all constraints. The corners of the feasible region, known as vertices, are of particular importance as they are potential candidates for yielding the optimal solution of the objective function.

Constraints

Constraints are specific conditions that the variables in your problem must satisfy. They help in defining the scope of your feasible region.

Our problem presents four distinct constraints:

• \( x \geq 0 \): This ensures that \( x \) cannot be negative.
• \( y \geq 0 \): Similarly, \( y \) is also non-negative.
• \( 3x + y \leq 15 \): This describes a linear inequality involving both \( x \) and \( y \). On a graph, it appears as a line dividing the plane, with the region below the line included in the feasible area.
• \( 4x + 3y \leq 30 \): Another inequality that creates another boundary on the graph.

Constraints can be equality or inequality expressions, defining where solutions can and cannot exist. Solving these involves using them to limit the space in which we search for the optimal value of the objective function.

Intersection Points

Intersection points are critical as they often represent potential solutions to optimization problems. In linear programming, these points form where the constraints intersect, typically at the corners of the feasible region.

To find these points, you can solve equations derived from the constraints simultaneously. For our problem, one key intersection is found by solving \( 3x + y = 15 \) and \( 4x + 3y = 30 \) together, resulting in the point \( (3,6) \).

• Intersection points are calculated by setting the equations equal and solving them as a system of equations.
• They help pinpoint where exactly the feasible region's boundaries meet.

After determining these intersection points, you can then substitute them into the objective function to evaluate potential outcomes, and ultimately find maximum or minimum values.