Problem 65 - College Algebra

Question

For the following exercises, solve for the indicated value, and graph the situation showing the solution point. An account with an initial deposit of \(\$ 6,500\) earns \(7.25 \%\) annual interest, compounded continuously. How much will the account be worth after 20 years?

Step-by-Step Solution

Answer

The account will be worth approximately $27,768.65 after 20 years.

Step 1: Understand Continuous Compounding Formula

The formula for continuously compounded interest is given by \( A = Pe^{rt} \), where \( A \) is the amount of money accumulated after \( t \) years, including interest. \( P \) is the principal amount (initial deposit), \( r \) is the annual interest rate (in decimal), and \( t \) is the time in years.

Step 2: Identify Given Values

From the problem, we know that the initial deposit (\( P \)) is 6500, the annual interest rate (\( r \)) is 7.25%, which in decimal form is 0.0725, and the time (\( t \)) is 20 years.

Step 3: Substitute Values into the Formula

Substitute the given values into the formula: \[ A = 6500 \cdot e^{0.0725 \times 20} \].

Step 4: Calculate the Exponential Factor

Compute the exponent: \( 0.0725 \times 20 = 1.45 \). Thus, the expression becomes \( A = 6500 \cdot e^{1.45} \).

Step 5: Evaluate the Expression

Use a calculator to find \( e^{1.45} \). This is approximately 4.2721. So, the equation becomes \( A = 6500 \times 4.2721 \).

Step 6: Final Calculation

Complete the multiplication: \( 6500 \times 4.2721 = 27768.65 \). Thus, the account will be worth approximately $27,768.65 after 20 years.

Key Concepts

Exponential Growth

Exponential growth is a key concept in continuously compounded interest. It describes a process where quantities grow by a consistent percentage over regular intervals. In the context of financial mathematics, it is used to model how money increases over time when it accrues interest.

Unlike linear growth, where the quantity increases by a constant amount, exponential growth means the amount increases by a fixed percentage. This percentage applies to an amount that itself is always increasing.

In the formula for continuously compounded interest, \( A = Pe^{rt} \), the part \( e^{rt} \) represents exponential growth. Here, \( P \) is the initial amount, \( r \) is the rate of growth, and \( t \) is time. As time progresses, the effect of the exponential factor \( e^{rt} \) makes the initial quantity \( P \) grow exponentially.

Interest Rate Calculation

Calculating interest rates is a fundamental part of understanding financial returns. When dealing with continuously compounded interest, the interest rate is expressed as a decimal rather than a percentage.

To convert a percentage to a decimal, simply divide by 100. For instance, an interest rate of 7.25% becomes 0.0725 in decimal form.

This decimal is then used in the exponential growth model. Breaking it down into steps:

• Start with the annual interest rate in percentage.
• Convert this percentage to a decimal by dividing by 100.
• Use this decimal in the formula \( A = Pe^{rt} \).

By calculating the interest rate in this way, you ensure accuracy in predicting future values based on continuously compounded interest.

Financial Mathematics

Financial mathematics involves various models and equations to understand how money grows over time. Continuous compounding is a powerful concept in this field, allowing for precise calculations of future value.

The formula \( A = Pe^{rt} \) is central to this type of calculation, representing the accumulation of interest over time with continuous compounding.
Financial mathematics uses such equations to:

• Project future investment growth.
• Evaluate the impact of different interest rates over time.
• Analyze the longevity and feasibility of financial plans.

By mastering these mathematical tools, individuals and businesses can make informed decisions about investments and savings.

Graphing Financial Equations

Graphing financial equations helps visualize how investments grow over time. In the case of continuously compounded interest, a graph can show exponential growth more clearly.

When you graph the equation \( A = Pe^{rt} \), you typically see a curve that starts slow and becomes steeper over time. This steepening illustrates the accelerating growth intrinsic to exponential functions.

Key points to consider when graphing:

• The x-axis represents time \( t \).
• The y-axis shows the total accumulated value \( A \).
• The graph starts at \( P \), the initial investment value, and rises exponentially.

By graphing these equations, students and investors can better grasp the implications of interest rates and time on financial growth.