Problem 81 - College Algebra with Modeling and Visualization

Question

Continuous Compounding Over 5 years, the total value of a mutual fund account decreases continuously by \(15 \%\). Find a formula \(A(x)\) that calculates the amount of money in the account after \(x\) years.

Step-by-Step Solution

Answer

The formula is \( A(x) = P \cdot e^{-0.15x} \).

Step 1: Understanding Continuous Compounding

In the case of continuous compounding, the formula we use to find the amount of money in an account over time is: \ A(x) = P \cdot e^{rx} \ where \(A(x)\) is the amount after \(x\) years, \(P\) is the initial principal balance, \(r\) is the rate of growth or decay (in decimal form), \(x\) is the number of years, and \(e\) is Euler's number, approximately equal to 2.71828.

Step 2: Determining the Decay Rate

The problem tells us the account decreases by 15% continuously over 5 years. This suggests a decay rate. To convert 15% to a decimal, we use \( r = -0.15 \). The negative sign indicates a decrease.

Step 3: Formulate the Function A(x)

Using the decay rate, we substitute \(r = -0.15\) into the continuous compounding formula. Hence, we have: \ A(x) = P \cdot e^{-0.15x}. \ This formula will give the amount in the account after \(x\) years.

Key Concepts

Continuous Compounding

Continuous compounding is a fundamental concept in finance that describes how interest adds to a principal balance in an infinitely frequent manner. Unlike annual, semi-annual, or monthly compounding, continuous compounding allows the interest to be calculated and added at every possible moment. This results in the accumulation of interest leading to exponential growth or decay of resources, depending on the nature of the rate (positive for growth, negative for decay).
This process can be expressed using the formula:

• \( A(x) = P \cdot e^{rx} \)

Where:

• \( A(x) \) is the amount after \( x \) years.
• \( P \) is the initial principal amount.
• \( r \) is the rate of interest or decay, converted to decimal form.
• \( x \) is the elapsed time in years.
• \( e \) is the mathematical constant, approximately 2.71828.

Continuous compounding is particularly useful in fields that require high precision in interest calculations, such as financial engineering and economic forecasting.

Decay Rate

The decay rate is a crucial aspect in understanding exponential decay, which is the process of reducing an amount by a consistent percentage over a set period of time. In our context, the decay rate is represented as a negative percentage, because it describes a decrease over time.
To convert a percentage into a decimal for use in formulas, divide by 100. For instance, a 15% decay rate becomes \( -0.15 \). The negative sign is essential as it indicates a decline rather than an increase.
In the formula \( A(x) = P \cdot e^{rx} \), the variable \( r \) stands for the decay rate:

• If \( r \) is negative, the scenario represents decay.
• If \( r \) is positive, it represents growth.

Decay rates are commonly used to describe various real-world decreases, such as depreciation of assets, decrease in population, or reduction in energy resources.

Exponential Functions

Exponential functions are a class of mathematical functions that describe scenarios where growth or decay happens at an exponential rate, meaning the rate of change is proportional to the current state. They can be recognized by their distinctive curve shapes, rapidly increasing or decreasing, depending on the sign of the exponent.
The general form of an exponential function is \( A(x) = P \cdot e^{rx} \), where:

• \( A(x) \) is the amount at time \( x \).
• \( P \) is the initial quantity.
• \( r \) determines the type of change (growth if positive, decay if negative).

These functions model a variety of real-life processes, such as population growth, radioactive decay, and financial investments when compounded continuously.
Exponential functions are powerful tools due to their ability to model both rapid growth and rapid decay, making them highly applicable in numerous scientific and financial analyses.