an investment triples every 12 years.\nwhat is the annual interest rate (compounded continuously)?\nround to…

Explanation:

Step1: Recall the continuous - compounding formula

The formula for continuous compounding is (A = P e^{rt}), where (A) is the final amount, (P) is the principal amount, (r) is the annual interest rate, and (t) is the time in years.

Step2: Substitute the given values into the formula

Given that the investment triples, so (A = 3P) and (t = 12). Substituting into (A=Pe^{rt}), we get (3P=Pe^{12r}).

Step3: Simplify the equation

Divide both sides of the equation (3P = Pe^{12r}) by (P) (since (P\neq0)). We obtain (3=e^{12r}).

Step4: Take the natural logarithm of both sides

Using the property (\ln(e^{x})=x), if (3 = e^{12r}), then (\ln(3)=\ln(e^{12r})). So (\ln(3)=12r).

Step5: Solve for (r)

We know that (\ln(3)\approx1.0986). Then (r=\frac{\ln(3)}{12}). Substituting (\ln(3)\approx1.0986) into the formula, (r=\frac{1.0986}{12}).

Step6: Calculate the value of (r)

(r=\frac{1.0986}{12}=0.0916)

Answer:

(r = 0.0916)