Problem 20 - Algebra 1: Concepts and Skills

Question

Write an exponential decay model for the investment. $550 is placed in a mutual fund. Then the value decreases by 4% per year.

Step-by-Step Solution

Answer

The exponential decay model for the investment is \(A = 550(1 - 0.04)^t\).

Step 1: Converting Percentage to Decimal

First, convert the percentage decrease to a decimal. 4% is 0.04 in decimal form.

Step 2: Formulating the Exponential Decay Model

Using the exponential decay equation, plug the given values into the formula. We have \(P = $550\) (the initial investment), \(r = 0.04\) (the decay rate), and \(t\) as the number of years, which we leave as a variable. Therefore the mathematical model for this problem is \(A = 550(1 - 0.04)^t\).

Key Concepts

Percentage to Decimal Conversion

Understanding how to convert a percentage into a decimal is crucial in many math problems, especially in handling exponential decay models. This process is straightforward.

• First, think of the percentage as being out of 100. This is why the word "percent" means "per 100."
• For example, 4% is equivalent to 4 out of 100.
• To convert this percentage to decimal form, you divide by 100. This means that 4% becomes 0.04 when written as a decimal.

Converting percentages to decimals makes it easier to use them in equations, such as the exponential decay formula. By converting accurately, you ensure your calculations will reflect the correct decay rate.

Exponential Decay Equation

The exponential decay equation represents processes where quantities decrease over time. It is described by the formula: \[ A = P(1 - r)^t \] where:

• \(A\) is the amount remaining after time \(t\).
• \(P\) is the initial amount (or principal).
• \(r\) is the decay rate, expressed as a decimal.
• \(t\) is the time that has passed.

In our problem, an initial investment of $550 decreases at a rate of 4% per year. By inserting \(P = 550\) and \(r = 0.04\) into the formula, we obtain: \[ A = 550(1 - 0.04)^t \]
This exponential decay equation illustrates how the investment value changes annually.

Mathematical Modeling

Mathematical modeling helps us understand and simulate real-world problems like finance, science, and engineering. When dealing with an investment that decreases in value over time, we apply an exponential decay model.

• First, identify the important components: the initial value, the rate of change, and how time influences the process.
• By creating a model, you can predict future outcomes based on current data.

For example, the investment problem involves creating a model to calculate future values of an investment as it decreases by 4% annually. By utilizing the exponential decay equation, you can predict how the value will diminish over time. This mathematical foresight is pivotal for financial planning and projections.