Problem 32 - Calculus Early Transcendentals: Pearson New International Edition

Question

If Methuselah's parents had put \(\$ 100\) in the bank for him at birth and he left it there, what would Methuselah have had at his death ( 969 years later) if interest was \(4 \%\) compounded annually?

Step-by-Step Solution

Answer

Methuselah would have about \( 5.1984 \times 10^{17} \) dollars.

Step 1: Identify Key Values

We need to identify the principle (initial investment), interest rate, and the total time period. The principle is \( P = 100 \), the interest rate is \( r = 0.04 \) (4% expressed as a decimal), and the time period is \( t = 969 \) years.

Step 2: Compound Interest Formula

The compound interest formula is used to calculate the future value of an investment. The formula is: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Where \( A \) is the amount, \( P \) is the principal, \( r \) is the annual interest rate, \( n \) is the number of times the interest is compounded per year, and \( t \) is the time in years. In this problem, \( n = 1 \) since the interest is compounded annually.

Step 3: Substitute Values into the Formula

Now, substitute the known values into the compound interest formula: \[ A = 100 \left(1 + \frac{0.04}{1}\right)^{1 \times 969} \] This simplifies to: \[ A = 100 (1.04)^{969} \]

Step 4: Calculate the Future Value

Use a calculator to compute \( (1.04)^{969} \). This is a very large number, approximated to \( \approx 5.1984 \times 10^{15} \). Thus, \( A = 100 \times 5.1984 \times 10^{15} = 5.1984 \times 10^{17} \).

Step 5: Conclusion

Methuselah would have had approximately \( 5.1984 \times 10^{17} \) dollars at his death, which is a number much larger than today's global GDP.

Key Concepts

Interest Rate

The interest rate is a crucial component of compound interest calculations. It represents the percentage of the principal amount that will be added to the investment over a specific period. In our scenario, an annual interest rate of 4% means that every year, 4% of the principal is added to the account.
This percentage is expressed as a decimal in mathematical calculations, so we convert 4% to 0.04 before using it in any formula.

• Importance: A higher interest rate leads to more rapid growth of your investment.
• Effect on Compound Interest: With compounding, the interest is not only earned on the principal amount but also on the accumulated interest from previous periods.

Understanding the interest rate helps clarify why certain investments grow faster than others. It's a key factor in the exponential growth of an investment.

Principal Amount

The principal amount, also known as the initial investment, is the starting sum of money put into savings or investment.
In Methuselah's case, his parents initially deposited $100 into a bank account. This principal acts as the foundation for future growth under the influence of compound interest.

• Base Value: The principal is the base on which interest is calculated. Larger principal amounts earn more interest.
• Initial Impact: At the beginning, the impact of interest might seem small, but over time, compounds significantly enhance the principal’s value.

The principal is critical because it represents the initial financial commitment, which ultimately determines the investment's potential over time.

Exponential Growth

Exponential growth occurs when the growth rate of a value is proportional to its current size, resulting in a constant percentage increase over time.
This is the magic behind compound interest.

• Ever-Increasing Rate: With each passing year, the interest is calculated on a growing amount, leading to a faster and faster increase in total value.
• Example: In Methuselah's account, due to the 4% interest compounded annually over 969 years, the initial $100 grows into an enormously large sum due to this exponential effect.

The power of exponential growth demonstrates how small, consistent interest rates over long periods can lead to substantial financial increases. Understanding this helps in comprehending why starting to invest or save early is beneficial.