Final answer:
To calculate the time required for an investment to double or triple at an interest rate of 4.15%, compounded continuously, use the formula A = Pe^(rt) and solve for 't.' It takes approximately 16.7 years to double and 26.7 years to triple the investment.
Step-by-step explanation:
To determine the time required for an investment to double or triple when compounded continuously, you can use the formula A = Pert, where A is the amount of money accumulated after n years, including interest, P is the principal amount (the initial amount of money), r is the annual interest rate (decimal), t is the time the money is invested for, and e is the base of the natural logarithm. Here, we want to find t when A is twice or three times the initial principal P with r = 0.0415.
Finding Time to Double
To double the investment, we set A to 2P, so the equation becomes:
2P = Pe0.0415t
Dividing both sides by P, we get:
2 = e0.0415t
Taking natural logarithms of both sides, we find:
ln(2) = 0.0415t
Therefore, the time required to double is:
t = ln(2) / 0.0415 ≈ 16.7 years
Finding Time to Triple
To triple the investment, we set A to 3P and follow a similar process:
3 = e0.0415t
ln(3) = 0.0415t
t = ln(3) / 0.0415 ≈ 26.7 years