Problem 29 - Calculus Early Transcendentals: Pearson New International Edition

Question

Manhattan Island is said to have been bought by Peter Minuit in 1626 for \(\$ 24\). Suppose that Minuit had instead put the \(\$ 24\) in the bank at \(6 \%\) interest compounded continuously. What would that \(\$ 24\) have been worth in 2000 ?

Step-by-Step Solution

Answer

Approximately \$124.7 billion in 2000.

Step 1: Understand the Method

Continuous compound interest is calculated using the formula \( A = P e^{rt} \), where \( A \) is the future amount, \( P \) is the principal amount, \( r \) is the interest rate, and \( t \) is the time in years.

Step 2: Identify Variables

From the question, we find that:- Principal \( P = 24 \)- Interest rate \( r = 0.06 \)- Time from 1626 to 2000, \( t = 2000 - 1626 = 374 \) years.

Step 3: Substitute Values into the Formula

Replace \( P \), \( r \), and \( t \) in the formula to get: \[ A = 24 \, e^{0.06 \times 374} \].

Step 4: Calculate the Exponent

Calculate \( 0.06 \times 374 = 22.44 \), so the formula becomes \( A = 24 \, e^{22.44} \).

Step 5: Calculate \( e^{22.44} \)

Using a calculator, compute \( e^{22.44} \). Using approximations, \( e^{22.44} \approx 5.196 \times 10^{9} \).

Step 6: Calculate the Future Value

Multiply \( 24 \) by the result from Step 5: \[ A = 24 \, \times \, 5.196 \times 10^{9} \approx 1.24704 \times 10^{11} \].

Step 7: Conclusion

Thus, the future value of the \( \\(24 \) would have been approximately \\)124,704,000,000 in the year 2000.

Key Concepts

Interest Rate

When discussing continuous compound interest, the interest rate is a crucial component. It's the percentage at which the initial investment grows over time. In our problem, the interest rate is given as 6%, or 0.06 when written as a decimal. Understanding how this rate impacts your money is fundamental.

• The interest rate directly influences how fast your money grows.
• Higher interest rates lead to more significant growth, while lower rates slow it down.
• The interest rate must be converted into a decimal before using it in exponential growth formulas.

For continuous compounding, this rate suggests that interest is applied and calculated at every possible moment. This results in a steady and smooth increase in your investment value over time.

Exponential Growth

Exponential growth is a powerful concept that describes how investments grow using compound interest. It means the amount of money you have increases at a rate proportional to its current value. In the context of continuous compounding, the formula used to represent this is: \[ A = P e^{rt} \] where:

• \( A \) is the amount of money accumulated after time \( t \), including interest.
• \( P \) is the principal or initial amount invested.
• \( e \) is the base of the natural logarithm, approximately equal to 2.71828.
• \( r \) is the interest rate expressed as a decimal.
• \( t \) is the time period in years.

Exponential growth occurs because as your investment earns interest, the interest itself earns interest. This compounding effect leads to a rapid increase in your investment, given enough time.

Future Value

Future value refers to the worth of an investment after a certain period, with interest applied. It's an essential part of financial planning and investment. In our example, it tells us how much Peter Minuit's initial \( \, \\( 24 \) would be worth in the year 2000. Using the formula \( A = P e^{rt} \), we find the investment's future value. Here’s how it works:

• Start with the principal amount, which is the initial investment.
• Apply the interest rate continuously over the given time period.
• The future value, \( A \), is found by calculating \( e^{rt} \), then multiplying by the principal.

In simple terms, the future value tells you how much your investment grows over time. It's not just about the initial deposit but also about how the interest accumulates. For instance, in our case, it demonstrates that even a small initial amount like \( \\) 24 \) can grow vastly with enough time and continuous compounding.