Problem 67 - College Algebra with Modeling and Visualization

Question

\(\$ 2000\) at \(10 \%\) compounded continuously for 8 years

Step-by-Step Solution

Answer

The amount after 8 years is approximately $4451.08.

Step 1: Understand Continuous Compounding

Continuous compounding means that interest is calculated and added to the account balance at every possible instant. The formula for continuous compounding is given by \( A = Pe^{rt} \), where \( A \) is the amount of money accumulated after n years, including interest, \( P \) is the principal amount (the initial amount of money), \( r \) is the annual interest rate (as a decimal), and \( t \) is the time the money is invested for in years.

Step 2: Identify the Known Values

From the problem statement, we have the following values: Principal \( P = 2000 \), interest rate \( r = 10\% = 0.10 \), and time \( t = 8 \) years. Substitute these values into the continuous compounding formula.

Step 3: Substitute Known Values into Equation

Using the continuous compounding formula \( A = Pe^{rt} \), substitute \( P = 2000 \), \( r = 0.10 \), and \( t = 8 \). The equation becomes \( A = 2000 \, e^{(0.10 \times 8)} \).

Step 4: Calculate the Exponent

Now, calculate the exponent, \( rt = 0.10 \times 8 = 0.8 \). Therefore, the equation becomes \( A = 2000 \, e^{0.8} \).

Step 5: Solve for the Exponential Part

Calculate \( e^{0.8} \) using a calculator. \( e^{0.8} \approx 2.22554 \).

Step 6: Calculate the Final Amount

Multiply the principal by the calculated exponential value: \( A = 2000 \, \times \, 2.22554 \). This gives \( A \approx 4451.08 \). Thus, the amount after 8 years is approximately $4451.08.

Key Concepts

Exponential Growth

Exponential growth is a fundamental concept in mathematics that describes how quantities increase rapidly over time. In the context of continuous compound interest, exponential growth occurs because the interest compounds continuously rather than at discrete intervals like annually or monthly. This means the interest calculation happens at every possible moment.

Unlike linear growth, where the increase is consistent, exponential growth accelerates as time goes on. This is depicted by the formula for continuous compounding: \[ A = Pe^{rt} \]- Here, the term \( e^{rt} \) represents the exponential growth factor.- \( e \) is Euler's number, approximately equal to 2.71828, a constant representing the base of natural logarithms.
In our example, the growth is powered by compounding 10% interest over 8 years. The critical takeaway is that as long as the principal is invested, the account grows faster, illustrating the power of exponential growth. This makes a significant difference in the accumulated amount over long periods.

Interest Rate Calculation

Understanding how the interest rate affects growth is crucial. In our problem, we're dealing with a 10% annual interest rate, which equals an interest rate of 0.10 when converted into a decimal format for calculations.
To determine how much the principal amount increases due to the interest rate, we need to consider:- The annual rate as a decimal: \( r = 0.10 \)- How it affects growth over time: the product \( rt = 0.10 \times 8 = 0.8 \)This multiplication is used in the exponential part of the formula \( Pe^{rt} \), demonstrating how the rate contributes to total growth. With continuous compounding, even a small change in the interest rate can lead to much larger accumulated amounts because of the exponential growth influence.
Moreover, continuous compounding amplifies this effect more than standard periodic compounding methods do, making it a powerful tool for maximizing investment returns.

Principal Amount

The principal amount is the initial sum of money invested or lent. It forms the baseline upon which interest calculations are performed. In our example, this is the \( $2000 \) initial investment.
When calculating continuous compound interest, the principal amount \( P \) is the starting point of the formula \( A = Pe^{rt} \). Understanding the role of the principal is essential because:- It is the base amount that grows over time.- The larger the principal, the more interest can be earned, assuming the same growth conditions (like interest rate and time).The principal is crucial because even small increments can ultimately lead to substantial variations in the final amount. Bigger principal balances benefit from exponential growth resulting from continuous compounding. In essence, the principal acts as the seed money, and its size influences the end result greatly.