For investments compounded continuously, we have
[tex] A = Pe^{rt} [/tex]
where P is the principal amount and A is the new amount after a period of time t.
Given the information that we have, we can solve for the time taken for the principal amount to reach $1000 as shown below.
[tex] 1000 = 600(e^{0.035(t)}) [/tex]
[tex]\frac{1000}{600} = e^{0.035t} [/tex]
For powers of e, we can use the natural logarithmic function, ln(). Recall that
[tex] x = ln(e^{x}) [/tex]
Using what we know, we can solve for t in our equation.
[tex] ln(\frac{1000}{600}) = ln(e^{0.035t}) [/tex]
[tex]ln(\frac{5}{3}) =0.035t [/tex]
[tex]t = \frac{ln(\frac{5}{3})}{0.035} [/tex]
[tex] t = 14.59 [/tex]
From this, we can see that it will take about 15 years for a principal amount of $600 to reach $1000 when compounded continuously at 3.5%.
Answer: 15 years
