Problem 19 - Algebra for College Students

Question

Use the formula \(A=P e^{n}\) to find the total amount of money accumulated at the end of the indicated time period by compounding continuously. \(\$ 2000\) for 15 years at \(10 \%\) \(\$ 8963.38\)

Step-by-Step Solution

Answer

The accumulated amount is approximately \$8963.40.

Step 1: Understanding the Formula

The formula given is \(A = P e^{rn}\). Here, \(A\) represents the total amount of money accumulated after time period \(n\) with principal \(P\), continuous compounding at rate \(r\). In our problem, \(P = 2000\), \(n = 15\) years, and the interest rate \(r = 0.10\) (10\%).

Step 2: Substitute Values into Formula

Substitute the known values into the formula \(A = P e^{rn}\). So, we have \(A = 2000 e^{0.10 \times 15}\). This becomes \(A = 2000 e^{1.5}\).

Step 3: Calculate \(e^{1.5}\)

Use a calculator to find \(e^{1.5}\). We approximate \(e^{1.5} \approx 4.4817\).

Step 4: Calculate the Total Amount \(A\)

Now, substitute \(e^{1.5}\) back into the equation to find \(A\). So, \(A \approx 2000 \times 4.4817\).

Step 5: Final Computation

Multiply the principal by \(e^{1.5}\). Thus, \(A \approx 2000 \times 4.4817 = 8963.40\).

Key Concepts

Continuous Compounding

Continuous compounding is a powerful concept in finance that allows money to grow at its most efficient rate. It assumes that interest is added to the principal constantly, leading to exponential growth. Continuous compounding uses the natural exponential function, where the basis is the mathematical constant \( e \). This constant is approximately equal to 2.71828. Here, the growth is always ongoing, meaning it doesn't just happen annually or semi-annually but at every possible moment. This results in money growing faster than with regular compounding intervals.
By using continuous compounding, investors and savers can maximize their returns over time, especially important in long-term investments. It highlights the appeal of leaving savings untouched to benefit fully from the accumulated interest.

Interest Rate Calculation

Interest rate calculation in continuous compounding involves determining how much interest a principal amount earns over time. The interest rate is expressed as a decimal in calculations. For example, an interest rate of 10% would be expressed as 0.10.
This calculation incorporates the time variable, such as the number of years the investment is held, referred to as \( n \) in the formula. The equation used for continuous compounding is:
\[ A = P e^{rn} \]
Here, \( A \) is the total accumulated amount, \( P \) is the principal amount, \( r \) is the interest rate, and \( n \) is the time period. The equation shows that money will grow exponentially, driven by the interaction between the time period and the interest rate. Understanding how to input these variables correctly into the formula is crucial for accurately calculating the future value of investments.

Mathematical Formulas

In the context of continuous compounding, mathematical formulas are essential for accurately predicting the future value of an investment. The key formula used in these calculations is \( A = P e^{rn} \). This formula is derived from the basic principle that money can grow exponentially when left to compound continuously.

• \( A \): Total amount after time \( n \)
• \( P \): Original principal
• \( e \): Euler's number, approximately 2.71828
• \( r \): Annual interest rate (as a decimal)
• \( n \): Number of years

The exponential function \( e^{rn} \) showcases growth as a curve that rapidly rises, making it much more significant over time due to the effect of continuously adding interest. By applying this formula accurately, individuals can plan investments and understand potential returns more tangibly, underscoring the importance of being comfortable with its use in financial contexts.