Problem 32 - Algebra and Trigonometry

Question

Compound Interest If \(\$ 3500\) is invested at an interest rate of 6.25\(\%\) per year, compounded continuously, find the value of the investment after the given number of years. \(\begin{array}{llll}{\text { (a) } 3 \text { years }} & {\text { (b) } 6 \text { years }} & {\text { (c) } 9 \text { years }}\end{array}\)

Step-by-Step Solution

Answer

After 3 years: $4224.96. After 6 years: $5100.25. After 9 years: $6157.66.

Step 1: Understand the Formula

For continuously compounded interest, we use the formula \( A = Pe^{rt} \), where \( A \) is the amount of money accumulated after \( n \) years, including interest, \( P \) is the principal amount (initial investment), \( r \) is the annual interest rate (in decimal), and \( t \) is the time in years.

Step 2: Convert Interest Rate

Convert the annual interest rate from a percentage to a decimal for calculation purposes. Thus, \( r = \frac{6.25}{100} = 0.0625 \).

Step 3: Calculate for 3 Years

Substitute \( P = 3500 \), \( r = 0.0625 \), and \( t = 3 \) into the formula \( A = 3500e^{0.0625 \times 3} \). Solve to find \( A \).

Step 4: Calculate for 6 Years

Substitute \( P = 3500 \), \( r = 0.0625 \), and \( t = 6 \) into the formula \( A = 3500e^{0.0625 \times 6} \). Solve to find \( A \).

Step 5: Calculate for 9 Years

Substitute \( P = 3500 \), \( r = 0.0625 \), and \( t = 9 \) into the formula \( A = 3500e^{0.0625 \times 9} \). Solve to find \( A \).

Step 6: Compute using Calculator

Using a calculator, compute the exponential expressions to find the values: (a) 3 years: \( A \approx 3500 \times e^{0.1875} \approx 4224.96 \) (b) 6 years: \( A \approx 3500 \times e^{0.375} \approx 5100.25 \) (c) 9 years: \( A \approx 3500 \times e^{0.5625} \approx 6157.66 \).

Key Concepts

Continuously Compounded Interest

Continuously compounded interest is a unique way to accrue interest by assuming that interest is added to the principal balance infinitely throughout the year. This means it grows slightly faster than your typical compounded interest, where additions occur at specific intervals like annually, semi-annually, or quarterly.
With continuous compounding, the formula used is \( A = Pe^{rt} \), where:

• \( A \) represents the final amount.
• \( P \) is the principal, or initial investment amount.
• \( r \) is the annual interest rate in decimal form.
• \( t \) denotes the time the money is invested for, measured in years.
• \( e \) is the base of the natural logarithm, approximately equal to 2.71828.

As a result of this infinite compounding, the calculations often result in a slightly higher amount compared to other compounding methods. This concept is deeply rooted in advanced mathematics involving exponential growth and can be a beneficial approach for long-term investments.

Exponential Growth

Exponential growth is a fundamental concept applied in various fields, from biology to finance. In the context of finance, it describes how investment value can increase rapidly over time due to interest compounding.
Essentially, exponential growth occurs when the growth rate is proportional to the current value, leading to the potential for significant increases as time continues.
When interest compounds continuously, it follows the exponential growth pattern, meaning your investment can grow faster because each portion of the added interest then earns interest.

• Think of exponential growth like a snowball growing as it rolls down a hill; it picks up more snow, and its mass increases faster as it grows larger.
• In real-world applications, recognizing and applying exponential growth can maximize your returns on investments, especially over lengthy periods.

Thus, understanding exponential growth helps investors make informed decisions, ensuring they maximize gains by leveraging the compounding effects effectively.

Investment Calculations

Investment calculations require a deep understanding of how different types of interest rates affect the growth of your initial money. For continuously compounded investments, the calculations are derived from the exponential formula \( A = Pe^{rt} \), which requires several steps.

• First, identify your initial principal amount (\( P \)). In this exercise, it's \( 3500 \) dollars.
• Convert the annual percentage rate into a decimal (e.g., 6.25% becomes 0.0625).
• Determine the product of the rate \( r \) and the time \( t \), which represents the power to which \( e \) is raised.
• Use a calculator to find \( e^{rt} \).
• Multiply the result by your principal \( P \) to get the future value \( A \).

Investment calculations involving continuous compounding tend to yield higher values over time compared to other compounding frequencies. Thus, mastering these calculations can provide substantial advantages in both planning and evaluating potential investments.