Continuously Compounded Interest

[theuniverse]

1. The problem:

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.

 

2. The attempt at the 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!

Edited September 30, 2010 by theuniverse

[Mr Skeptic]

Continuously compounded at a yearly rate is not the same as compounded yearly. The $500 you put in in January will earn you more interest than the $500 you add in November.

 

Try it with calculating and adding in the interest from the previous amount each time you add your monthly savings.

[theuniverse]

I'm having troubles coming up with the formula.. so what do you think about that: S(t) = S(initial)*e^rt + (500 + 5t).

I use S(initial)*e^rt to describe the change to the initial amount, plus (500 + 5t) for the additional savings.

Edit: since t is in years I think (500 + 5t) should actually be (6000+60t)...

Edited September 30, 2010 by theuniverse