Problem 29 - Applied Calculus

Question

The island of Manhattan was sold for $$\$ 24$$ in \(1626 .\) Suppose the money had been invested in an account which compounded interest continuously. (a) How much money would be in the account in the year 2005 if the yearly interest rate was (i) \(5 \% ?\) (ii) \(7 \%\) ? (b) If the yearly interest rate was \(6 \%\), in what year would the account be worth one million dollars?

Step-by-Step Solution

Answer

(i) $2.32 \times 10^{10}$ dollars; (ii) $1.23 \times 10^{14}$ dollars; (b) the year 2042.

Step 1: Understand Continuous Compounding Formula

Continuous compounding of interest is calculated using the formula \( A = P e^{rt} \), where \(A\) is the amount of money accumulated after time \(t\), \(P\) is the principal amount (initial investment), \(r\) is the interest rate (as a decimal), and \(t\) is the time in years.

Step 2: Calculate the Time Elapsed

Determine the number of years between 1626 and 2005. This difference gives us the time \(t\). \(t = 2005 - 1626 = 379\) years.

Step 3: Calculate Accumulated Amount at 5% Interest

Using the continuous compounding formula, plug in the values for \(P = 24\), \(r = 0.05\), and \(t = 379\): \[ A = 24 \times e^{0.05 \times 379} \] Calculate this to find \(A \approx 2.32 \times 10^{10}\) dollars.

Step 4: Calculate Accumulated Amount at 7% Interest

Repeat the calculation with \(r = 0.07\): \[ A = 24 \times e^{0.07 \times 379} \] Calculate this to find \(A \approx 1.23 \times 10^{14}\) dollars.

Step 5: Set Up Equation for Future Value at 6% Interest

For part (b), we want to find the time \(t\) when \(A\) reaches one million dollars with an interest rate of \(6\%\). Set up the equation: \[ 1000000 = 24 \times e^{0.06 \times t} \]

Step 6: Solve for Time Needed to Reach One Million Dollars

Solve the equation from Step 5 for \(t\): 1. Divide both sides by 24: \[ e^{0.06 \times t} = \frac{1000000}{24} \]2. Take the natural logarithm of both sides: \[ 0.06 \times t = \ln\left(\frac{1000000}{24}\right) \]3. Divide by 0.06: \[ t = \frac{\ln\left(\frac{1000000}{24}\right)}{0.06} \] Calculate this to find \(t \approx 415.6\) years. Therefore, the year would be approximately \(1626 + 416 = 2042\).

Key Concepts

Compound Interest

Compound interest is like a snowball effect on your money. Instead of earning interest only on your initial amount, known as the principal, compound interest allows you to earn interest on the interest already accumulated. This means your balance grows faster over time!
Imagine putting away some money in a savings account. Not only does the bank pay you interest on your initial amount, but as this interest is added to your balance, it also starts earning interest.

• The magic of compound interest lies in its ability to grow wealth quickly over time.
• It depends on the frequency of compounding - daily, monthly, or in this case, continuously.
• This means more compounding periods result in more interest.

With continuous compounding, you take this concept to the extreme, as interest is calculated and added an infinite number of times in a year! This can seem a bit mind-blowing, but mathematically, it smooths out to a specific formula that makes calculations easy.

Exponential Growth

Exponential growth is a rapid increase in number due to the continuous and compounded accumulation. It's the essence of why compound interest has such a powerful effect over time. Instead of a straight line or a curve that grows slowly, with exponential growth, the curve starts slow but then skyrockets upwards at a certain point.
In our context, exponential growth describes how the value of an investment grows over time when interest compounds continuously.
This is captured by the mathematical expression:\[ A = P e^{rt} \]where:

• \(P\) is the principal or initial amount,
• \(r\) is the interest rate, and
• \(t\) is the time.

Exponentially growing functions are important because they can help predict not just investments, but also many real-world phenomena like population growth or radioactive decay.
Understanding the concept helps you realize the amazing potential of time in growing investments!

Interest Rates

The interest rate is a crucial piece of the compound interest and exponential growth puzzle. It represents the percentage of your principal that you earn as interest over a specific period.
When it comes to continuous compounding, the interest rate is particularly important because a small increase can significantly influence the final amount.

• A higher interest rate means more money earned over the same period of time.
• This rate is usually expressed as an annual percentage.
• Even small changes in the interest rate can greatly affect the exponential growth of your investment. For instance, moving from a 5% to a 7% rate led to a massive difference in the future values in our example.

It's essential to understand that small differences add up extraordinarily in the long run. Being savvy about interest rates can help you select investment opportunities that align with your financial goals.