Problem 27 - Calculus and its applications

Question

Peter Minuit of the Dutch West India Company purchased Manhattan Island from the natives living there in 1626 for \(\$ 24\) worth of merchandise. Assuming an exponential rate of inflation of \(5 \%\), how much will Manhattan be worth in \(2020 ?\)

Step-by-Step Solution

Answer

Manhattan's worth in 2020 would be approximately \(2.775 \times 10^{21}\) dollars.

Step 1: Understanding the Problem

We need to calculate the value of Manhattan Island in 2020, given that it was bought in 1626 for $24. We'll use exponential growth to determine its worth accounting for an average annual inflation rate of 5% over the time period from 1626 to 2020.

Step 2: Formula for Exponential Growth

The formula for exponential growth is: \( FV = PV \times (1 + r)^n \)Where: - \( FV \) is the future value (Manhattan's worth in 2020), - \( PV \) is the present value (\$24 in 1626), - \( r \) is the rate of inflation (5% or 0.05), and - \( n \) is the number of years between 1626 and 2020.

Step 3: Determine the Number of Years

First, calculate the number of years between 1626 and 2020. \( n = 2020 - 1626 \). Thus, \( n = 394 \) years.

Step 4: Apply the Exponential Growth Formula

Now, substitute the values into the formula:\[ FV = 24 \times (1 + 0.05)^{394} \]Calculate the result to find the worth of Manhattan in 2020.

Step 5: Calculation Result

After performing the calculation:\[ FV = 24 \times (1.05)^{394} \approx 2.775 \times 10^{21} \]Thus, the value of Manhattan in 2020 would be approximately \(2.775 \times 10^{21}\) dollars due to the effects of compound interest over 394 years.

Key Concepts

Inflation Rate

Inflation rate is a term used to describe the percentage increase in the price level of goods and services over a period of time. It is an essential concept in economics because it affects purchasing power and economic decision-making. When we talk about a 5% inflation rate, it means that, on average, prices increase by 5% per year.

Understanding inflation allows us to make informed decisions about investments and savings. If inflation is higher than the return on your savings, the real value of your money decreases. For instance, if Manhattan was adjusted for a regular inflation rate of 5%, its value has grown exponentially over time.

Inflation impacts various aspects of life, including:

• Buying power: Your money buys less than it did in the previous period.
• Investment strategy: Investors seek returns that outpace inflation.
• Salary negotiations: Employees may ask for raises to keep up with the cost of living.

Future Value Calculation

The idea of future value calculation is determining how much a current asset will be worth at a future date, given a specific rate of return or inflation. This calculation is crucial for individuals making long-term investments or assessing changes in the value of money.

In simple terms, future value tells us how much something you own today will grow over a time period at a certain growth rate. The formula used for this is:

\[ FV = PV \times (1 + r)^n \]

Where:

• \(FV\) is the future value
• \(PV\) is the present value or initial investment
• \(r\) is the annual growth rate or inflation rate
• \(n\) is the number of years in the future you are evaluating

By applying this formula to our Manhattan example, we calculate that an investment of $24 in 1626, growing at a 5% annual rate for 394 years, results in the astronomical future value we computed.

Compound Interest

Compound interest refers to the process where interest is added to the principal amount of a deposit or loan. This added interest then earns interest on itself in subsequent periods. That's why it's called 'compounding'; it is the interest on interest.

Compound interest is a powerful financial concept because it can significantly increase the total amount of money earned or owed over time. It differs from simple interest, which is only calculated on the principal amount.

To visualize compound interest, think of it as a snowball rolling down a snowy hill. As the snowball rolls, it gathers more snow. With each roll, the amount of snow collected increases, which in turn makes the snowball grow larger more quickly.

For calculating the future value using compound interest, we use the same exponential growth formula. This principle was used in the step by step calculation for determining Manhattan's worth in 2020. By applying the interest over 394 years, the small initial amount grows into an incredibly large sum due to the power of compounding.