Problem 36 - Finite Mathematics and Applied Calculus

Question

Enormous State University's Business School is buying computers. The school has two models to choose from: the Pomegranate and the iZac. Each Pomegranate comes with \(400 \mathrm{MB}\) of memory and \(80 \mathrm{~GB}\) of disk space, and each iZac has 300 MB of memory and 100 GB of disk space. For reasons related to its accreditation, the school would like to be able to say that it has a total of at least \(48,000 \mathrm{MB}\) of memory and at least \(12,800 \mathrm{~GB}\) of disk space. Draw the feasible region that shows the number of each kind of computer it can buy. Find the corner points of the region.

Step-by-Step Solution

Answer

The corner points of the feasible region for the number of Pomegranate (\(x\)) and iZac (\(y\)) computers can be found by graphing the constraints: \(400x + 300y \ge 48,000\), \(80x + 100y \ge 12,800\), \(x \ge 0\), and \(y \ge 0\). After graphing and finding the intersection points, we get the following corner points: Point A (solving \( \frac{48,000 - 400x}{300} = \frac{12,800 - 80x}{100} \)), Point B (setting \(x = 0\) in Memory constraint), Point C (setting \(x = 0\) in Disk space constraint), Point D (setting \(y = 0\) in Memory constraint), and Point E (setting \(y = 0\) in Disk space constraint).

Step 1: Define variables

Let \(x\) be the number of Pomegranate computers and \(y\) be the number of iZac computers.

Step 2: Write inequalities for the constraints

Based on the given constraints, we can set the following inequalities: 1. Memory constraint: \(400x + 300y \ge 48,000\) 2. Disk space constraint: \(80x + 100y \ge 12,800\) 3. Non-negative constraint: \(x \ge 0\) and \(y \ge 0\)

Step 3: Graph the inequalities

We can rewrite the constraints as follows to start graphing the inequalities: 1. Memory constraint: \(y \ge \frac{48,000 - 400x}{300}\) 2. Disk space constraint: \(y \ge \frac{12,800 - 80x}{100}\) Plot these lines on a graph, and shade the region that satisfies all constraints.

Step 4: Find the corner points

The feasible region is the intersection of all constraints. The boundary of this region is formed by the lines corresponding to constraints, and its corner points can be found by solving the equations of the lines. 1. Intersection of Memory constraint and Disk space constraint: \( \frac{48,000 - 400x}{300} = \frac{12,800 - 80x}{100} \) Solve for \(x\) and \(y\). Let this point be Point A. 2. Intersection of Memory constraint and Non-negative constraint on x-axis: Set \(x = 0\) in Memory constraint and solve for \(y\). Let this point be Point B. 3. Intersection of Disk space constraint and Non-negative constraint on x-axis: Set \(x = 0\) in Disk space constraint and solve for \(y\). Let this point be Point C. 4. Intersection of Memory constraint and Non-negative constraint on y-axis: Set \(y = 0\) in Memory constraint and solve for \(x\). Let this point be Point D. 5. Intersection of Disk space constraint and Non-negative constraint on y-axis: Set \(y = 0\) in Disk space constraint and solve for \(x\). Let this point be Point E. The corner points of the feasible region are Points A, B, C, D and E.

Key Concepts

Feasible Region

In linear programming, the concept of a feasible region is critical. It represents all possible solutions that satisfy a given set of constraints. Imagine it as a shaded area on a graph, where each point within this area represents a combination of decisions that meet all the required conditions.

For our exercise with the Pomegranate and iZac computers, the feasible region is determined by the constraints on memory and disk space, as well as the condition that the number of computers cannot be negative.

The equations given, such as the memory constraint \(400x + 300y \geq 48,000\) and disk space constraint \(80x + 100y \geq 12,800\), define the boundaries of this region. Graphically, these inequalities form lines, and the intersection of these lines determines the region of feasible solutions.

By graphing the inequalities, you can visually interpret which combinations of Pomegranate and iZac computers meet the school's requirements. Look for the intersecting lines on the graph and identify the shaded region.

• The shaded area is the feasible region, containing all the acceptable solutions.
• Only the combinations of computers within this region satisfy all constraints.

Understanding this region helps in decision-making by providing a visual and mathematical method to identify possible solutions.

Inequalities

Inequalities are foundational in linear programming as they define the constraints that mold the feasible region. In simple terms, inequalities express the limits of resources or restrictions within a problem. For this exercise, these inequalities relate to the memory and disk space requirements of purchasing computers.

To construct these mathematical expressions, break down the problem's conditions into inequalities:

• The memory constraint is represented by the inequality \(400x + 300y \geq 48,000\). This ensures the total memory from the computers is at least 48,000 MB.
• The disk space constraint is \(80x + 100y \geq 12,800\), guaranteeing a minimum of 12,800 GB in total disk space.
• Non-negativity ensures that you can't have a negative count of computers: \(x \geq 0\) and \(y \geq 0\).

These inequalities combine to outline the permissible set of solutions. By graphing these inequalities, you define a feasible region that represents all potential combinations of computers that meet the specified criteria.

This visual representation makes it easier to determine what combinations are possible and aids in further analysis, such as identifying the best solution for a specific objective.

Corner Points

Corner points are essential in linear programming because they often contain the optimal solution. These are the points where two or more constraint lines intersect within the feasible region. In this exercise, the goal is to identify these corner points as potential solutions.

To find the corner points, follow these steps:

• Identify where the lines represented by constraints intersect. These intersections are your potential corner points.
• Solve the equations of the intersecting lines to pinpoint exact coordinates for these points. This involves setting the expressions for two constraints equal and solving for the variables \(x\) and \(y\).

In our scenario:

• Point A is found by solving where the memory constraint and disk space constraint intersect.
• Point B emerges when the memory constraint intersects the x-axis, meaning set \(x = 0\) and solve for \(y\).
• Point C is where the disk space constraint intersects the x-axis (again, \(x = 0\)).
• Points D and E involve similar processes for intersections with the y-axis (\(y = 0\)).

Evaluating these points helps determine which combination of computers satisfies all constraints while possibly optimizing other criteria, like cost or resource use. In many optimization scenarios, the best solution lies at one of these corner points.