Problem 18 - 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). $$ \begin{aligned} 2 x+4 y & \geq 12 \\ x & \leq 5 \\ y & \leq 3 \\ x \geq 0, y & \geq 0 \end{aligned} $$

Step-by-Step Solution

Answer

The region corresponding to the given inequalities is a bounded triangle with corner points at (0, 3), (5, 3), and (5, 0).

Step 1: Write the inequalities in slope-intercept form

First, we need to rewrite each inequality in slope-intercept form (y = mx + b), then we can graph the lines. \( 2x + 4y \geq 12 \Rightarrow y \geq -\frac{1}{2}x + 3 \) \( x \leq 5 \) (Note that for vertical lines, we don't need to rewrite it as a slope-intercept form.) \( y \leq 3 \) \( x \geq 0, y \geq 0 \) (These are the x and y-axes.)

Step 2: Plot each inequality on the graph

Now, we plot each inequality on the graph and shade the region they encompass: 1. For the inequality \( y \geq -\frac{1}{2}x + 3 \), draw a solid line because it's a "greater than or equal to" inequality. Shade above the line, which represents values of y larger than the line. 2. For \(x \leq 5\), plot a vertical line at x=5 and shade left because x is "less than or equal to" 5. 3. For \(y \leq 3\), plot a horizontal line at y = 3 and shade below the line, representing the values of y "less than or equal to" 3. 4. Since the region is in the first quadrant, we will be only considering the region where both x and y are non-negative.

Step 3: Identify the corner points

Corner points are the points where the lines intersect within the shaded region. To find these points, we will solve the systems of equations formed by pairs of intersecting lines. Intersection 1: Line \(2x+4y=12\) with the y-axis (x=0): \( 2(0) + 4y = 12 \Rightarrow y = 3 \) The intersection point is (0, 3). Intersection 2: Line \(2x+4y=12\) with the x-axis (y=0): \( 2x + 4(0) = 12 \Rightarrow x = 6 \) This point (6, 0) is outside the inequality \( x \leq 5 \), so it is not a corner point. Intersection 3: Line \(y = 3\) with the x-axis (x=5): The intersection point is (5, 3). Intersection 4: Line \(x = 5\) with the y-axis (y=0): The intersection point is (5, 0).

Step 4: Determine whether the region is bounded or unbounded

A region is bounded if it has a finite area; otherwise, it is unbounded. In our case, the shaded area is a triangle with vertices at (0, 3), (5, 3), and (5, 0). It is enclosed from all sides and has a finite area. Therefore, the region is bounded.

Step 5: Summarize the final result

The region corresponding to the given inequalities is a triangle, which has a finite area, and it is bounded. The corner points of the region are (0, 3), (5, 3), and (5, 0).