I don't know much about money (I don't have so much of it) but your problem seems to be about exponential growth:
Control: Annually compounded at 3.5%:
Usually the continuously compounded capital is slightly higher than the annually compounded capital. My result contradict this rule because I rounded down ln(1+0.035) to 0.0344.
to b) I'm not sure what you are asking here, so this is only a try:
to c) Use the equation of a), plug in t = 2 and calculate:
to d) Now A(t) = 5000 $ and you have to solve for t:
. Divide by 4000 and logarithmize both sides:
. Solve for t. I've got
to e) Use the equation of b): Rate of change if A(t) = 5000 $ is A'(t) = 0.0344*5000 = 172 $ per year. (Remember: The rate of change at the beginning was 140 $/year. So now the capital is growing much faster)
to f and g) I guess that you want to know when the account reaches 12,000 $(?) Have a look at the answers to d) and e).
I've got the result: 31.936 years and 412.80 $/year.