Problem 5 - Precalculus: Functions and Graphs

Question

If \(P\) dollars is deposited in a savings account that pays interest at a rate of \(r \%\) per year compounded continuously, find the balance after \(t\) years. $$P=1000, \quad r=8 \frac{1}{4}, \quad t=5$$

Step-by-Step Solution

Answer

The balance after 5 years is approximately $1510.70.

Step 1: Understanding the Problem

To find the future balance of an account with continuous compounding interest, we use the formula \( A = Pe^{rt} \), where \( A \) is the final amount, \( P \) is the principal amount, \( r \) is the interest rate in decimal form, \( t \) is the time in years, and \( e \) is the base of the natural logarithm approximately equal to 2.71828.

Step 2: Convert Interest Rate to Decimal

The interest rate given is \( r = 8 \frac{1}{4}\,\% \). First, convert this percentage to a decimal form. \( 8 \frac{1}{4}\,\% = 8.25\,\% = \frac{8.25}{100} = 0.0825 \).

Step 3: Plug Values into Formula

Now, substitute the values \( P = 1000 \), \( r = 0.0825 \), and \( t = 5 \) into the formula \( A = Pe^{rt} \). This gives us: \( A = 1000 \cdot e^{0.0825 \times 5} \).

Step 4: Calculate the Exponent

Compute the exponent \( rt = 0.0825 \times 5 \). This is equal to \( 0.4125 \).

Step 5: Evaluate the Exponential

Find the value of \( e^{0.4125} \), which is approximately \( 1.5107 \).

Step 6: Calculate the Final Amount

Use the calculated value of \( e^{0.4125} \) to find \( A \). \( A = 1000 \times 1.5107 \approx 1510.70 \).

Step 7: Solution Interpretation

The final balance of the account after 5 years, given the continuous compounding interest rate, is approximately \( \$1510.70 \).

Key Concepts

Exponential Growth

Exponential growth refers to a process where the rate of increase is proportional to the current size. This concept is crucial in understanding continuous compounding. Continuous compounding is calculated using the formula \( A = Pe^{rt} \), where \( A \) is the amount after time \( t \), \( P \) is the principal, \( r \) is the annual interest rate in decimal form, and \( e \) is the base of natural logarithms (approximately 2.71828). The formula involves an exponential term \( e^{rt} \), showing how the growth of the savings balance accelerates over time.
In our example, consider the term \( e^{0.4125} \). This value illustrates how much the principal grows due to continuous compounding over a 5-year period. The key takeaway is that in exponential growth, as time increases, the accumulation of interest causes the total to rise faster and faster. This is different from simple or even standard compound interest, making continuous compounding a powerful tool for maximizing investment growth.

Interest Rate Conversion

Interest rate conversion involves changing a percentage, like 8.25%, into its decimal form, which is done for ease of calculation in continuous compounding formulas. For example, converting 8.25% to a decimal involves dividing by 100, resulting in 0.0825.
To apply the continuous compounding formula, it’s crucial to work in decimals rather than percentages. This allows for straightforward substitution into the formula \( A = Pe^{rt} \). Correct conversion ensures that the interest calculation correctly reflects the actual growth rate of the principal amount.
When tackling problems involving continuous compounding, always remember to convert any given percentage rates into decimals to guarantee accurate calculations. Missing this step can lead to significant miscalculations and a misunderstanding of the potential growth your investment or savings might achieve.

Natural Logarithm

Natural logarithms are an essential mathematical concept when dealing with continuous compounding. The natural logarithm has a base \( e \), which is approximately 2.71828. This constant arises naturally in many areas of mathematics, particularly in growth and decay processes.
In exponential functions like \( e^{rt} \), \( e \) serves as the foundation for computing how quantities grow continuously over time. Understanding \( e \) and its properties is essential in interpreting exponential growth. For example, in our exercise, calculating \( e^{0.4125} \) shows how much the principal has grown over time under a continuous compounding rate.
Utilizing natural logarithms in finance allows us to calculate growth and compounding processes efficiently. When working with continuous compounding, understanding how natural logarithms influence the formula can deepen comprehension and improve accuracy in calculating future values.