Question

Sketch the region determined by the constraints. Then find the minimum anc maximum values of the objective function and where they occur, subject to the indicated constraints. Objective function: $$ z=6 x+10 y $$ Constraints: $$ \begin{aligned} x & \geq 0 \\ y & \geq 0 \\ 3 x+5 y & \leq 15 \end{aligned} $$

Short answer

The maximum value of the objective function is 30 at points (0,3) and (5,0) and the minimum value is 0 at point (0,0).

Step-by-step solution

Sketch the Feasible Region

The constraints are in the form of inequalities indicating that the feasible region is a subset of the first quadrant (since \(x \geq 0\) and \(y \geq 0\)). To find this region, set each inequality to an equality and graph the lines. The line for the equation \(3x + 5y = 15\) passes through the points (0,3) and (5,0). The area below the line including the line itself in the first quadrant represents the inequality \(3x + 5y \leq 15\). The feasible region is therefore the area below the line in the first quadrant.

Finding the Corner Points

We know that, for a linear programming problem, the maximum and minimum of the objective function occur at corner points of the feasible region. In this case, the corner points are at (0,0), (0,3) and (5,0).

Evaluating the Objective Function

Now, evaluate the function \(z = 6x + 10y\) at each corner point. At (0,0), \(z = 0\). At (0,3), \(z = 30\). At (5,0), \(z = 30\).

Determining the Maximum and Minimum

Out of all values, the maximum value of z is 30 at points (0,3) and (5,0) and the minimum value is 0 at point (0,0).

Key concepts

Feasible Region

When working on a linear programming optimization problem, we first map out an area called the feasible region. This area contains all possible solutions to the problem that satisfy the given constraints. Think of the feasible region like a playground where all the points, representing solutions, are allowed to play. In our exercise, since both variables, x and y, must be non-negative, our playground is limited to the first quadrant of the graph, where both x and y are at or above zero. The inequality 3x + 5y ≤ 15 further shapes the playground, creating a triangular boundary within the first quadrant. This triangular zone entails every combination of x and y that simultaneously satisfies all the constraints, and therefore, delineates the feasible region for potential optimal solutions.

Objective Function

The objective function is the heart of a linear programming problem. It's the formula that we want to optimize - to make as large or as small as possible, based on the problem's goal. For our problem, the objective function is given by z = 6x + 10y. Our task is to either maximize or minimize z by finding the right mix of x and y values that fall within our feasible region. The objective function transforms our feasible region into a treasure map, showing where we might find the highest or lowest values of z, which are, in this case, the 'treasures' we're hunting for.

Inequality Constraints

Like the walls of a container, inequality constraints define the boundaries of where we can go in a linear programming problem. They consist of inequalities that all feasible solutions must satisfy. For this exercise, the constraints x ≥ 0 and y ≥ 0 confine us to the first quadrant, while the constraint 3x + 5y ≤ 15 forms an additional boundary line that contributes to shaping the feasible region. Picturing constraint equations as physical lines or barriers on a graph can help students visualize the limits within which they can operate, ultimately carving out the stage where the objective function will be performed.

Corner Point Method

To snag the optimal solution in a linear programming problem, a nifty technique called the corner point method comes into play. This method relies on the property that in a linear programming problem, the maximum or minimum value of the objective function occurs at one of the 'corners' or vertices of the feasible region. To use this method, locate the corner points of the feasible region—these are the points of intersection between the constraints. Once you've identified these points, you simply evaluate the objective function at each corner. By comparing these values, you can determine which corner offers the maximum or the minimum value of the objective function. In our case, the corner points are where the magic happens and are crucial for pinning down the bounds for z.

Optimal Solution

After the legwork of determining our feasible region and evaluating our objective function, we arrive at the climax of our linear programming quest: finding the optimal solution. This is essentially the 'best' solution according to the objective function, lying within the feasible region and respecting all constraints. In this exercise, by examining the objective function values at the corner points, we identify where z hits its maximum and minimum. In our example, we find the optimal solution to be where z equals 30, occurring at two corners, (0,3) and (5,0). This means these points are where we achieve the highest value of z, given our problems' constraints and goal. It's like hitting the jackpot in the treasure hunt of linear programming!