Final answer:
To find how long it will take for an investment to triple when continuously compounded at 15% per year, we can use the formula for compound interest: A = P * e^(rt), where A is the final amount, P is the principal (initial investment), e is the base of natural logarithms, r is the interest rate, and t is the time in years. Using this formula and solving the equation, we find that it will take approximately 9 years to triple the investment.
Step-by-step explanation:
To find how long it will take for an investment to triple when continuously compounded at 15% per year, we can use the formula for compound interest: A = P * e^(rt), where A is the final amount, P is the principal (initial investment), e is the base of natural logarithms, r is the interest rate, and t is the time in years. Let's set the final amount A to 3 times the principal, the initial investment. So we have:
3P = P * e^(0.15t)
Divide both sides by P:
3 = e^(0.15t)
To solve for t, we need to take the natural logarithm of both sides:
ln(3) = 0.15t * ln(e)
Since ln(e) is equal to 1, we can simplify:
ln(3) = 0.15t
Divide both sides by 0.15:
t = ln(3) / 0.15
Using a calculator, we can find that t ≈ 9.87 (rounded to the nearest year). Therefore, it will take approximately 9 years to triple the investment when continuously compounded at 15% per year.