Problem 67 - Precalculus with Limits

Question

Investment Analysis A person plans to invest up to \(\$ 20,000\) in two different interest-bearing accounts. Each account is to contain at least \(\$ 5000 .\) Moreover, the amount in one account should be at least twice the amount in the other account. Find and graph a system of inequalities to describe the various amounts that can be deposited in each account.

Step-by-Step Solution

Answer

The system of inequalities required to represent the given constraints is \(x \geq 5000, y \geq 5000, x + y \leq 20000, x \geq 2y\), or \(y \geq 2x\). The exact feasible region for both \(x\) and \(y\) will be determined by graphing these inequalities and determining where they all hold true.

Step 1: Defining the Variables

Let \(x\) represent the amount of money in the first account and \(y\) represent the amount in the second account. The amounts are given in dollars.

Step 2: Formulating the Inequalities

From the given conditions, we know that each account should contain at least $5000. This gives us our first two inequalities: \(x \geq 5000\) and \(y \geq 5000\). Moreover, the total investment is limited to $20000, giving another inequality: \(x + y \leq 20000\). The last condition stated that one account should have at least twice the amount in the other account. This results in two inequalities: \(x \geq 2y\) and \(y \geq 2x\). It's important to remember that these conditions might not all hold at the same time, creating a complex system.

Step 3: Graphing the System of Inequalities

The inequalities from Step 2 must now be graphed. The feasible region (representing valid amounts for \(x\) and \(y\)) will be where all inequality conditions are satisfied at the same time. This involves drawing each inequality on the same set of axes and identifying the region where all conditions are true. Because the created graphical representation will depend on the chosen order of inequalities and the chosen method for visualizing them, it is essential to be mindful of the requirement that all conditions must hold simultaneously.

Key Concepts

Investment Analysis Mathematics

The mathematics behind investment analysis involves using inequalities to represent conditions and constraints in financial decision-making. In our exercise, an individual is deciding how to allocate a maximum of \(\$20,000\) between two different interest-bearing accounts, each requiring a minimum deposit of \(\$5,000\).

In terms of investment analysis mathematics, translating real-life situations into mathematical inequalities allows for a structured approach to determining the feasible set of choices. This is critical because it ensures all investment regulations, such as minimum investment amounts and total capital limits, are adhered to. By mapping out inequalities, investors can visualize the range of possible allocations and make an informed decision that aligns with their financial strategy and goals.

Graphing Inequalities

Graphing inequalities is a key part of visualizing solutions to problems like the investment scenario presented. It allows us to comprehend the relationships between variables under certain constraints. When graphing inequalities such as \(x \geq 5000\) and \(y \geq 2x\), each inequality is represented as a region on a graph.

To graph the system, we begin by plotting each inequality on a two-dimensional coordinate system with \(x\) and \(y\) axes. For instance, the inequality \(x \geq 5000\) would be represented by a shaded region to the right of the line \(x = 5000\). The key is to find the overlap or intersection of these shaded regions since this represents all the pairs of \(x\) and \(y\) that satisfy all inequalities simultaneously. This intersection is known as the 'feasible region' and is of particular importance as it signifies all the possible combinations of investments in the two accounts that meet the person's constraints.

Visualizing Constraints

By utilizing coordinates, shading, and boundary lines, graphing makes abstract conditions tangible, helping us to understand the set of potential solutions useful in investment strategy planning.

Linear Programming

Linear programming is a method used to find the best outcome in a mathematical model whose requirements are represented by linear relationships. This problem-solving technique is fundamental in the field of optimization, where the goal is to maximize or minimize a linear function subjected to a set of linear inequalities, known as constraints.

In the context of our example, linear programming would help determine the optimal distribution of funds between the two accounts while respecting the constraints, such as the total amount to be invested and the minimum investment in each account. Simply put, it could tell us the best way to allocate the \(\$20,000\) to maximize interest earned.

Optimal Investment Strategy

By setting up a linear programming problem, we define the objective function, say, to maximize interest income, and apply the constraints given by our inequalities. Then, using methods such as the Simplex algorithm or graphical analysis, we identify the optimal solution that lies at a vertex, or corner point, of the feasible region on the graph. This point represents the best possible strategy for our investor within the given conditions.