Problem 20 - Finite Mathematics and Applied Calculus

Question

Sketch the region that corresponds to the given inequalities, say whether the region is bounded or unbounded, and find the coordinates of all corner points (if any). $$ \begin{aligned} &30 x+20 y \leq 600 \\ &10 x+40 y \leq 400 \\ &20 x+30 y \leq 450 \\ &x \geq 0, y \geq 0 \end{aligned} $$

Step-by-Step Solution

Answer

The region corresponding to the given inequalities is a bounded quadrilateral with corner points at (10, 7.5), (15, 7.5), (5, 8.75), and (0, 10).

Step 1: Graph the inequalities individually

To graph each inequality, we will first transform it into an equation and plot the line corresponding to the equation. Then we will determine which side of the line to shade based on the inequality sign. 1. \(30x+20y=600\), divide by 10: \(3x+2y=60\), so \(y=30-\frac{3}{2}x\). 2. \(10x+40y=400\), divide by 10: \(x+4y=40\), so \(y=10-\frac{1}{4}x\). 3. \(20x+30y=450\), divide by 10: \(2x+3y=45\), so \(y=15-\frac{2}{3}x\). 4. \(x\geq0\), the inequality corresponds to the vertical line \(x=0\). 5. \(y\geq0\), the inequality corresponds to the horizontal line \(y=0\).

Step 2: Graph the intersection region and determine whether it is bounded or unbounded

Now, we will graph the lines from Step 1 on the same coordinate plane, and find the region where all inequalities are satisfied. Since all inequalities are inequalities with less than or equal to (\(\leq\)), the region will be the intersection of the areas below the lines corresponding to each inequality, and also within the first quadrant as given by the conditions \(x\geq0\) and \(y\geq0\). By doing this, we can find the intersection region and determine that the region is bounded. The bounded region is a quadrilateral.

Step 3: Identify the corner points

Now that we have the bounded region, we will identify its vertices which are the corner points. We will find the intersection points of the lines from Step 1 that form the boundary of the region. There are four corner points where the lines intersect: 1. Intersection of \(3x+2y=60\) and \(x+4y=40\): Solving the system of equations, we get \(x=10, y=7.5\), so the point is (10, 7.5). 2. Intersection of \(3x+2y=60\) and \(2x+3y=45\): Solving the system of equations, we get \(x=15, y=7.5\), so the point is (15, 7.5). 3. Intersection of \(2x+3y=45\) and \(x+4y=40\): Solving the system of equations, we get \(x=5, y=8.75\), so the point is (5, 8.75). 4. Intersection of \(x+4y=40\) and \(x=0\): As \(x=0\), substituting in the equation, we get \(y=10\), so the point is (0, 10). The corner points of the region are (10, 7.5), (15, 7.5), (5, 8.75), and (0, 10).

Key Concepts

System of Inequalities

When dealing with more than one inequality involving the same set of variables, we call it a system of inequalities. Just like a system of equations, the solution to a system of inequalities is the set of all possible values that satisfy all the inequalities in the system. To solve graphically, each inequality is represented as a separate line, and for each, we shade the region that satisfies the inequality. When combined on the same graph, the overlapping shaded region becomes the visual solution to the system.

Imagine drawing a boundary around that region; any point inside that boundary is a solution to the system of inequalities. The reason for shading is to illustrate all the solutions for each inequality, thereby allowing us to easily identify where these solutions overlap, which is our primary area of interest. To ensure understanding, it's crucial to start by plotting the lines that represent the equality part of each inequality, then shading appropriately based on the inequality sign.

Bounded Region

A region represented on a graph is considered bounded if it's enclosed by lines or curves, essentially creating a 'fenced-in' area with no open ends. This means there are limits to the values that x and y can take within the system of inequalities. Conversely, an unbounded region would continue indefinitely in one or more directions. In the context of our exercise, the intersection of all shaded areas from each inequality results in a closed shape within the first quadrant, indicative of a bounded region. It's identified because each line forming the boundary of this region intersects with another line, rather than extending to infinity. Understanding whether a region is bounded is important as it can impact the optimization problems, where often the maximum or minimum values occur at the boundaries of such regions.

Corner Points

Corner points, also known as 'vertices', are the points at which the boundary lines of a bounded region intersect. These points are of particular interest in linear programming because, according to the extreme point theorem, if a linear function has a maximum or minimum value on a bounded region, it will occur at a corner point. To find these points, one must solve the systems of equations that correspond to the intersecting lines. For the upcoming test, focus on accurately graphing the lines and finding their intersections precisely. Once the corner points of the bounded region are identified, as in our exercise, they can be checked for the solutions to optimization problems or simply noted as the defining points of the solution set.

In problems that are not strictly theoretical, these points can represent practical, real-world limits like budgetary constraints or resource limits, where the intersections define the feasible options available.