Step 1 :We are given that the principal amount (P) is $1000, the amount of money accumulated after n years, including interest (A) is $3000, and the annual interest rate (r) is 5.7% or 0.057 in decimal form. We need to find the time the money is invested for in years (t).
Step 2 :The formula for continuous compounding interest is \(A = P e^{rt}\).
Step 3 :We can rearrange this formula to solve for t, giving us \(t = \frac{1}{r} \ln(\frac{A}{P})\).
Step 4 :Substituting the given values into this formula, we get \(t = \frac{1}{0.057} \ln(\frac{3000}{1000})\).
Step 5 :Solving this equation gives us \(t \approx 19\).
Step 6 :So, it will take approximately \(\boxed{19}\) years for the investment to triple.