The formula for continuous compounding is:
A = Pe^(rt)
where:
A = the amount of money at the end of the investment period
P = the principal amount (initial investment)
e = the mathematical constant approximately equal to 2.71828
r = the annual interest rate
t = the time in years
We are given that P = $5000, r = 3.5%, and we want to find t when A = $12,000. Substituting these values into the formula, we get:
$12,000 = $5000e^(0.035t)
Dividing both sides by $5000, we get:
2.4 = e^(0.035t)
Taking the natural logarithm of both sides, we get:
ln(2.4) = ln(e^(0.035t))
Using the rule of logarithms that ln(e^x) = x, we can simplify the right side to:
ln(2.4) = 0.035t
Dividing both sides by 0.035, we get:
t = ln(2.4) / 0.035 ≈ 26.7 years
Therefore, it will take approximately 26.7 years for the initial investment of $5000 to grow to $12,000 with continuous compounding at a rate of 3.5% per year.