Problem 16 - College Algebra and Calculus: An Applied Approach

Question

The simple interest received from an investment is directly proportional to the amount of the investment. By investing $$\$ 5000$$ in a municipal bond, you obtain interest of $$\$ 337.50$$ at the end of 1 year. Find a mathematical model that gives the interest \(I\) at the end of 1 year in terms of the amount invested \(P\).

Step-by-Step Solution

Answer

The mathematical model that gives the interest \(I\) at the end of 1 year in terms of the amount invested \(P\) is \(I = 0.0675P\).

Step 1: Establish the Proportional Relationship

Use the formula for simple interest which is \(I = kP\), where \(I\) is the interest, \(P\) is the principal (or investment), and \(k\) is the constant of proportionality.

Step 2: Substitute the Known Values

Substitute the values given in the problem into the formula. \(I\) is \$337.50 and \(P\) is \$5000, thus we have \$337.50 = k*\$5000.

Step 3: Solve for the Proportional Constant

Now, divide both sides of the equation by \$5000 to isolate \(k\). Doing this, we find that \(k = 0.0675\).

Step 4: Formulate the Mathematical Model

Now the mathematical model for the interest at the end of 1 year in terms of the amount invested can be written as \(I = 0.0675P\). This model suggests that for every dollar invested, interest of \$0.0675 will be received after one year.

Key Concepts

Proportional Relationship

When we talk about a proportional relationship, we're saying that two variables grow or shrink together at a constant rate. In the case of simple interest, the interest earned, represented as \(I\), is directly proportional to the amount you initially invest, known as the principal \(P\). This means that as the principal increases or decreases, the interest does so by the same factor.

By establishing this relationship, you can predict how much interest will accrue by changing the principal amount. Think of it like painting a wall: if you use more paint, the area you cover increases proportionally. Similarly, with investments, the more you invest, the more interest you'll earn, given the same rate of interest.

Simple Interest Formula

The formula used to calculate simple interest is fundamental in financial mathematics. It is given by the equation \(I = kP\). Here, \(I\) represents the interest, \(P\) is the principal, and \(k\) is the constant of proportionality, often understood as the interest rate.

This formula helps anyone with a basic understanding to easily compute the interest they can earn from a given principal amount over a specific period, usually a year. It assumes that the interest rate \(k\) is fixed and that both the principal \(P\) and time \(t\) are constants during the period. Using this formula can quickly reveal how different investment amounts will yield different interest returns.

Mathematical Modeling

Mathematical modeling is the process of creating a mathematical representation of a real-world situation. By using simple equations, like the simple interest formula, we model real-world financial situations to predict outcomes and make informed decisions.

In the context of the problem, once we found the proportional constant \(k = 0.0675\), it allowed us to formulate the equation \(I = 0.0675P\). This model accurately describes the relationship between any amount invested and the interest earned after one year.

Such models are crucial for making investment decisions and planning financial strategies. They simplify complex processes and help in drawing clear conclusions without the need for intricate calculations or computer simulations.

Investment Principal

The investment principal, denoted as \(P\), is the original sum of money that you invest or lend. It is the starting block for calculating interest and is pivotal in the world of finance.

In our simple interest calculation, the principal amount \(P\) is the driver's seat—without it, no interest can be generated. By understanding your principal, you'd know exactly how much money is being "put to work" to earn interest.

• This forms the basis for future interest calculations.
• The larger the principal, the greater the expected returns, assuming the interest rate stays constant.

Comprehending the concept of principal can empower students and investors to make sound financial choices, laying the foundation for wealth accumulation.