Problem 78 - Intermediate Algebra

Question

Assume that there are no deposits or withdrawals. Determining the Previous Balance. An account now contains \(\$ 3,610\) and has been accumulating interest at \(8 \%\) annual interest, compounded continuously. How much was in the account 4 years ago?

Step-by-Step Solution

Answer

The account had approximately \$2,621.62 four years ago.

Step 1: Understanding Continuous Compounding

With continuous compounding, the formula to calculate the future amount is given by \[ A = Pe^{rt} \] where \( A \) is the future value, \( P \) is the principal amount (initial balance), \( r \) is the interest rate as a decimal, and \( t \) is the time in years. Our goal is to find the initial amount \( P \).

Step 2: Identifying Known Values

We know the following values from the problem: - \( A = 3610 \) (the amount now in the account),- \( r = 0.08 \) (8% interest rate),- \( t = 4 \) years. We need to find \( P \), the initial amount.

Step 3: Rearrange the Formula to Solve for P

Rearrange the continuous compounding formula to solve for \( P \):\[ P = \frac{A}{e^{rt}} \]

Step 4: Calculating the Exponential Term

Calculate the exponential term \( e^{rt} \):First, calculate the product of \( r \) and \( t \):\( rt = 0.08 \times 4 = 0.32 \).Then calculate \( e^{0.32} \) using a calculator, which gives approximately 1.377127764.

Step 5: Calculate the Initial Amount P

Divide the current balance \( A \) by the calculated exponential term to find \( P \):\[ P = \frac{3610}{1.377127764} \]Perform the division: \( P \approx 2621.62 \).

Key Concepts

Exponential Growth

Exponential growth refers to a process where the quantity increases over time at a rate proportional to its current value. In financial contexts, this process is often related to the way investments grow thanks to interest accrual. When interest is compounded continuously, the growth is modeled by the exponential function \ \( e^{rt} \), where \ \( r \) is the interest rate and \ \( t \) is the time period. This results in the balance curve sharply rising upward as time progresses, reflecting the power of exponential growth.

• Continuous compounding provides a constant growth rate.
• It highlights how money grows faster each period.
• The growth rate is exponential, not linear.

To visualize exponential growth in financial terms: if you invested, your money grows not just on the initial capital but also on the interest already earned. This accumulating nature fuels the exponential curve, providing more rapid returns compared to simple interest.

Interest Rate Calculation

Calculating the interest on an account is fundamental in financial mathematics. For continuous compounding, it's crucial to understand how the constant \ \( e \) plays a key role. The formula for future value when compounded continuously is: \ \( A = Pe^{rt} \).
To work out how much was initially invested (the principal \ \( P \)), you rearrange the formula: \ \( P = \frac{A}{e^{rt}} \). This involves determining an exponential term, \ \( e^{rt} \), to scale down the current account value back to its original amount at the start of the period.

• Convert the annual interest rate percentage to a decimal.
• Multiply this decimal by the time in years to find \ \( rt \).
• Compute the exponential term using \ \( e^{rt} \).

Once you have these values, you can effectively calculate the principal. This process emphasizes understanding both basic arithmetic and the properties of exponential functions.

Financial Mathematics

Financial mathematics is the backbone of all modern financial analysis and planning. It involves using mathematical formulas and models to solve financial problems, such as calculating loan payments, investment growth, and account balances. The application of continuous compounding is a shining example of financial mathematics at work.

• Models like \ \( A = Pe^{rt} \) help predict future values.
• They assist in comparing different investment options.
• Financial math simplifies complex economic phenomena.

When dealing with interest rates and investment timings, financial mathematics allows individuals to make informed decisions about their money. By grasping these fundamental concepts, such as continuous compounding and exponential growth, one can better understand the tools and strategies financial professionals use every day.