Problem 54 - College Algebra with Modeling and Visualization

Question

Find the maximum value of \(P=3 x+5 y\) subject to the following constraints. $$ \begin{array}{r} 3 x+y \leq 8 \\ x+3 y \leq 8 \\ x \geq 0, y \geq 0 \end{array} $$

Step-by-Step Solution

Answer

The maximum value is 16 at the point (2, 2).

Step 1: Identify Constraints

The constraints for the problem are \(3x + y \leq 8\), \(x + 3y \leq 8\), \(x \geq 0\), and \(y \geq 0\). These inequalities form a feasible region where the values of \(x\) and \(y\) must lie.

Step 2: Graph the Constraints

Plot each inequality on the coordinate plane. The lines \(3x + y = 8\) and \(x + 3y = 8\) are boundaries for the feasible region, along with the axes \(x = 0\) and \(y = 0\). Identify the intersection of these half-planes, which forms a polygon.

Step 3: Determine Corner Points

Find the intersection points (corner points) of the lines bounding the feasible region. Solving \(3x + y = 8\) and \(x + 3y = 8\) gives the point \((2, 2)\). Intersection of \(3x + y = 8\) with \(x=0\) gives \((0, 8)\), and with \(y=0\) gives \((\frac{8}{3}, 0)\). Intersection of \(x + 3y = 8\) with \(x=0\) gives \((0, \frac{8}{3})\), and with \(y=0\) gives \((8, 0)\). Only feasible points are \((0, \frac{8}{3})\), \((2, 2)\), and \((\frac{8}{3}, 0)\).

Step 4: Evaluate Objective Function at Corner Points

Substitute each corner point into the objective function \(P = 3x + 5y\). For \((0, \frac{8}{3})\): \(P = 5 \times \frac{8}{3} = \frac{40}{3}\) = 13.33. For \((2, 2)\): \(P = 3\times2 + 5\times2 = 6 + 10 = 16\). For \((\frac{8}{3}, 0)\): \(P = 3\times\frac{8}{3} = 8\).

Step 5: Identify the Maximum Value

Compare the values obtained for the objective function at each corner point. The maximum value of \(P\) occurs at the point \((2,2)\), where \(P = 16\).

Key Concepts

Feasible Region

In linear programming, the feasible region is a crucial concept that refers to the set of all possible solutions that satisfy the given constraints of the problem. It's like a map on a coordinate grid, showing where solutions live. This region is determined by the inequalities, each of which creates a boundary in the system.
For the given exercise, we have four constraints forming the feasible region:

• The inequality \(3x + y \leq 8\) forms one boundary.
• The inequality \(x + 3y \leq 8\) forms another boundary.
• Additionally, \(x \geq 0\) and \(y \geq 0\) ensure that the feasible region is limited to the first quadrant.

Where these boundaries intersect, inside the shaded area, is the feasible region. This polygonal shape is where we will search for the solution to the objective function.

Objective Function

The objective function in a linear programming problem is what you aim to optimize — either maximize or minimize. This function is expressed as a linear equation of the decision variables.
In this case, the objective function is represented by \(P = 3x + 5y\). Here, \(P\) stands for the value you want to maximize. The ultimate goal is to find the combination of \(x\) and \(y\) that will yield the highest value of \(P\), while sitting in the feasible region.
The coefficients in this function, 3 and 5, indicate the contribution of each variable to the value of \(P\). Thus, the task is to compute \(P\) at different points within our feasible region and identify the maximum value.

Constraints

Constraints in a linear programming problem define the limitations or restrictions on the decision variables. They are written as linear inequalities.
For our problem, the constraints given are:

• \(3x + y \leq 8\)
• \(x + 3y \leq 8\)
• \(x \geq 0\) and \(y \geq 0\)

These constraints confine our solution to a feasible region on the graph. Each inequality slices the plane into half, creating boundaries for where \(x\) and \(y\) can lie.
Working with these constraints, you can pinpoint the possible solutions by finding intersection points of these boundary lines, which help us form the feasible region.

Graphical Method

The graphical method is a visual approach to solving linear programming problems. By plotting the constraints as lines on a graph, you can visually interpret how these constraints interact with each other.
Here's how it's done:

• Draw each constraint as a line on the coordinate plane.
• Shade the region that satisfies each inequality. The intersection area of all shaded regions is your feasible region.
• Identify the corner points of this feasible region, as they are potential solutions.

For this exercise, the critical points were found where the constraints intersect, creating corners at points like \((2,2)\), \((0, \frac{8}{3})\), and \((\frac{8}{3}, 0)\).
By applying the objective function to these points, you determine which one yields the maximum value, thus finding the optimal solution through a straightforward visual process.