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

The \$24 would be worth about \$125.44 billion in 2000.

Step 1: Identify the Formula

To find the future value of an investment with continuously compounded interest, use the formula \( A = Pe^{rt} \), where \( A \) is the amount in the future, \( P \) is the initial principal balance (\$24), \( r \) is the rate of interest (6% or 0.06), and \( t \) is the time in years.

Step 2: Calculate the Time Period

Calculate the duration from 1626 to 2000. It is 2000 - 1626 = 374 years. This represents the number of years the initial amount is compounded.

Step 3: Substitute the Values into the Formula

Substitute the known values into the formula: \( P = 24 \), \( r = 0.06 \), and \( t = 374 \), so the equation is \( A = 24e^{0.06 \, \times \, 374} \).

Step 4: Calculate the Exponent

Calculate the exponent part: \( 0.06 \times 374 = 22.44 \). This is the exponent for \( e \).

Step 5: Compute the Future Value

Calculate the value of \( e^{22.44} \), then multiply by 24 to get \( A \). Using a calculator, \( e^{22.44} \approx 5.2265 \times 10^9 \). Now compute \( 24 \times 5.2265 \times 10^9 \approx 1.2544 \times 10^{11} \). Therefore, the investment would be worth approximately \$125.44 billion.

Key Concepts

Future Value Calculation

When you're thinking about how much an investment might grow, the future value tells you how much it will be worth after some time. In the world of finance, the future value is key. It lets you know the potential worth of your money if you invest it now. For calculations involving continuously compounded interest, the formula used is \[A = Pe^{rt}\]

• \( A \) is the future value of the investment.
• \( P \) represents the initial amount of money you start with.
• \( r \) is your annual interest rate.
• \( t \) is time, measured in years.

The neat part about using this formula is how it continuously calculates the interest. Think of it as your money working non-stop, without taking breaks!

Interest Rate

Whenever you save or borrow money, you will encounter interest rates; they are everywhere! An interest rate tells you how much the amount will either grow (in case of savings) or shrink (in case of loans) over time. Interest rates are expressed as a percentage of the original amount. There are different types of interest rates:

• Simple Interest: Calculated once a year on the initial amount.
• Compound Interest: Adds interest on top of interest - it compounds! It can be annually, quarterly, monthly, etc.

But if you want your money growth to be even quicker, there's continuous compounding. It's like compound interest on speed. With continuous compounding, your interest is calculated at every moment.

Exponential Growth

Exponential growth sounds complex, but it's an exciting idea. Imagine a small snowball rolling down a hill. As it rolls, it picks up more snow, grows larger, and speeds up. Similarly, exponential growth isn't just regular increase; it's a growth rate that compounds constantly. This concept applies to continuously compounded interest, where your money keeps growing as long as it sits in the bank. Every tiny bit of interest you earn starts earning its own interest, leading to rapid and substantial increase over time. In math terms, exponential growth is modeled with the equation:\[y = P e^{rt}\]Here, - \( y \) is your growing amount over time,- \( P \) is your starting point,- \( e \) is the base of the natural logarithm (approximately 2.718),- \( r \) is the growth rate,- and \( t \) is time.With exponential growth, just like the snowball, your investment can become quite large if given enough time.