# DE - Continuously Compounded Interest

• theuniverse
In summary: Your Name]In summary, we have a savings account with an initial balance of zero and a continuous savings rate of $500 per month, increasing by$5 every month. The account also earns continuously compounded interest at a rate of 8% per year. To estimate how long it will take to save $1 million, we can use the formula for future value of an annuity, taking into account the increasing savings rate. theuniverse ## Homework Statement A new savings account with an initial balance of zero is made. You save money continuously, at a rate of$500 per month. Also, every month you plan to increase this rate by $5. you've found a bank account that pays continously compounded interest at a rate of 8% per year. Estimate how long it will take for you to save one million dollars. ## Homework Equations (dS/dt)= rS+k ## The Attempt at a Solution I decided to take care of the saving rate first: since my interest is per year I decided to convert the savings to a yearly rate as well where k=500*12=6000, then I had to take care of the increments so I write it as 6000+60t where 60 is found by 5*12, and the t is in years so that every year 60$ are added to the initial saving rate.

I then tried to use the formula: S(t) = S(initial)*e^rt + [(k+60t)/r][(e^rt)-1)]
I subbed in my values, and 1 million for S(t) but the problem is that I can't isolate for t and always end up having e^0.08t - t = some number.

Is there a different way I should approach this problem?
Thanks!

Thank you for your post. It looks like you have made a good attempt at solving this problem. However, there are a few errors in your approach that may be causing the difficulty in isolating for t.

Firstly, the formula you have used, S(t) = S(initial)*e^rt + [(k+60t)/r][(e^rt)-1)], is not applicable in this case. This formula is used for continuously compounded interest with a constant interest rate. In this problem, the interest rate is not constant as it increases every month.

Secondly, you have converted the monthly savings rate to a yearly rate, which is correct. However, you have not taken into account the increasing rate of $5 per month. This means that your savings rate should actually be 500+5t, where t is in years. To solve this problem, you can use the formula for future value of an annuity, which is FV = Pmt*[((1+r)^n - 1)/r], where FV is the future value, Pmt is the periodic payment, r is the interest rate per period, and n is the number of periods. In this case, the future value you are trying to reach is$1 million, the periodic payment is 500+5t, the interest rate per period is 8%/12=0.0067, and the number of periods is the number of months it will take to reach \$1 million. You can then solve for t using algebra and taking the natural logarithm of both sides.

I hope this helps and good luck with your calculations!

## 1. What is continuously compounded interest?

Continuously compounded interest is a type of interest calculation where the interest earned on an investment is added to the principal amount continuously, rather than at regular intervals. This results in a compounding effect, where the interest is earned on both the initial principal and the accumulated interest.

## 2. How is the continuously compounded interest formula derived?

The continuously compounded interest formula is derived from the mathematical concept of exponential growth, where the growth rate is continuously applied to the initial value. In finance, this formula is used to calculate the future value of an investment with continuous compounding.

## 3. What is the difference between continuously compounded interest and simple interest?

The main difference between continuously compounded interest and simple interest is the frequency at which the interest is calculated and added to the principal. With simple interest, the interest is only calculated and added at regular intervals, while with continuous compounding, it is added continuously. This results in a higher overall return on investment with continuous compounding.

## 4. How does the frequency of compounding affect the value of an investment?

The frequency of compounding has a direct impact on the value of an investment. The more frequently the interest is compounded, the higher the overall return will be. This is because with more frequent compounding, the interest is added to the principal more often, resulting in a compounding effect.

## 5. What are some real-world applications of continuously compounded interest?

Continuously compounded interest is commonly used in finance and banking, particularly in investments such as bonds, mutual funds, and savings accounts. It is also used in the calculation of loan interest and mortgage payments. Additionally, it is used in population growth models and other mathematical applications.

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