Final answer:
Using the formula for continuous compounding, it will approximately take 22.12 years for a $3000 investment at an annual interest rate of 5% compounded continuously to triple.
Step-by-step explanation:
To determine how long it will take for an investment of $3000 to triple at an annual interest rate of 5% compounded continuously, we use the formula for continuous compounding, which is A = Pert, where A is the amount of money accumulated after n years, including interest, P is the principal amount, r is the annual interest rate, and t is the time in years.
To triple the investment, we want to find t when A is three times the principal amount, P. Thus, we are looking for t in the equation 9000 = 3000e0.05t. Taking natural logarithms on both sides of the equation, we get ln(9000/3000) = ln(e0.05t), which simplifies to ln(3) = 0.05t. Solving for t, we get t = ln(3) / 0.05. Therefore, it will take approximately 22.12 years for the investment to triple.