1. ## Continuously Compounded Interest

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!

2. you only have to give an approximate answer so you dont ahve to solve for t. You can use trial and error to find approximate values of t (say, to the nearest month).

NB: Ive assumed your formula for S(t) is correct and haven't checked it.