Problem 31 - 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

The \$24 would have been worth about \$128 billion in 2000.

Step 1: Identify the Formula

For continuous compounding interest, we use the formula \( A = Pe^{rt} \). Here, \( A \) is the future value, \( P \) is the principal amount, \( r \) is the annual interest rate, and \( t \) is the time in years.

Step 2: Set Initial Parameters

Given in the problem: \( P = 24 \), \( r = 0.06 \) (since 6% is written as a decimal), and \( t = 2000 - 1626 = 374 \) years.

Step 3: Calculate the Future Value

Substitute the known values into the formula to find \( A \):\[A = 24e^{0.06 \times 374}\]First, calculate the exponent:\[0.06 \times 374 = 22.44\]Then, calculate \( e^{22.44} \).

Step 4: Compute Exponential Result

Use a calculator to find \( e^{22.44} \approx 5.337 imes 10^{9} \).

Step 5: Final Calculation

Now, multiply this by the principal amount:\[A = 24 imes 5.337 imes 10^{9} = 1.28088 imes 10^{11}\]Therefore, the future value of the \\(24 investment is approximately \( \\)1.28 \times 10^{11} \).

Step 6: Interpret Result

This result means that \\(24 continuously compounded at 6% interest from 1626 to 2000 would grow to a massive \\)128 billion.

Key Concepts

Future Value Calculation

Calculating the future value of an investment, especially with continuous compounding, is a fascinating concept that showcases the power of exponential growth over time. The formula used for continuous compounding is given by \( A = Pe^{rt} \). Here:

• \( A \) is the future value of the investment.
• \( P \) refers to the principal, which is the initial amount invested.
• \( r \) stands for the annual interest rate, expressed as a decimal.
• \( t \) is the time period in years.

When you plug values into this formula, you can determine how much an investment will grow over time with continuous interest accrual. In our original problem, Peter Minuit's $24 deposit would compound every moment, leading to an enormous amount due to the ongoing effect of exponential growth.

Exponential Growth

Exponential growth is a concept where quantities increase at a rate proportional to their current value. This means that as the value grows, the rate of growth accelerates. This is common in contexts where compounding occurs, such as in finance and population studies. With continuous compounding, the interest is added to the principal so frequently that it can be thought of as growing instantly every moment. In the exercise example, the initial $24, when compounded at a 6% continuous rate from 1626 to 2000, results in a gigantic amount due to this exponential nature. Over 374 years, the rate of growth allows the investment's worth to balloon significantly, illustrating how powerful compounding can be over long periods. An understanding of exponential growth helps highlight why investments tend to expand so dramatically over extended durations.

Historical Investment Analysis

Analyzing historical investments, such as the hypothetical compounding of Peter Minuit's $24, provides valuable insights into financial growth potential over long time horizons. This kind of analysis helps to compare how investments might perform over centuries based on different compounding rates. Historically, small initial investments can lead to substantial future values when left to grow over significant periods. For example, an investment with continuous compounding, optimally maintained in a bank, would amass considerable value despite the initial low principal. This exercise teaches not only the principle of compounding but also the significance of allowing investments to grow uninterrupted over time. By comparing the historical context with modern scenarios, students can better appreciate the magnitude and potential of sustained financial growth.