DE - Continuously Compounded Interest

[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!

 

[mighty2000]

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!