Continuously compunded interest

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In summary, the conversation discusses the sale of the island of Manhattan for $24 in 1626 and the potential growth of that money if it had been invested in an account with continuous compound interest. The person asking the question shares their calculations for different interest rates and the other person confirms that the numbers are correct, despite being very large due to exponential growth over a long period of 379 years.
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Homework Statement


The island of Manhattan was sold for $24 in 1626. Suppose the money had been invested in an account which compounded interest continually.

a) How much money would be in the account in the year 2005 if the yearly interest rate was:
i: 5%? ii: 7%

The Attempt at a Solution



I put the numbers into the function P0ert and got

i:24e.05(379) and ii:24e.07(379)

but when I put that in my calculator I get VERY large numbers:

i: 4074662794 and ii: 7.980752573E12

For some reason, I don't think the homework answers would be such ridiculously large numbers. Did I do this right?
 
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  • #2
The numbers are ridiculously large but that doesn't mean they are wrong. Exponential growth is fast and 379 years is a ridiculously long time. I think you are correct.
 

1. What is continuously compounded interest?

Continuously compounded interest is a type of interest calculation where the initial principal amount, along with any accumulated interest, earns interest continuously over time. This means that the interest earned is immediately added to the principal, and then the new total amount continues to earn interest.

2. How is continuously compounded interest different from simple interest?

Simple interest is calculated as a percentage of the principal amount, while continuously compounded interest is calculated as a percentage of the total amount (principal + accumulated interest). This means that continuously compounded interest earns more interest over time compared to simple interest.

3. What is the formula for calculating continuously compounded interest?

The formula for calculating continuously compounded interest is A = P*e^(rt), where A is the final amount, P is the principal amount, e is the base of the natural logarithm, r is the annual interest rate (as a decimal), and t is the time period in years. This formula assumes that interest is compounded continuously throughout the time period.

4. How does the frequency of compounding affect continuously compounded interest?

The more frequently interest is compounded, the higher the interest earned will be. This is because with more frequent compounding, the interest is added to the principal more often, resulting in a higher total amount to earn interest on. In continuously compounded interest, the compounding is happening constantly, resulting in the highest possible interest earned.

5. What is the benefit of using continuously compounded interest?

The benefit of using continuously compounded interest is that it results in the highest possible interest earned compared to other compounding methods. This can be useful for long-term investments or loans, as the interest earned can significantly increase over time. It also simplifies the calculation of interest, as it only requires one formula regardless of the compounding frequency.

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