$600 were deposited into an account with a 5% interest rate, compounded continuously. how many years was it…

Explanation:

Step1: Write the continuous - compounding formula

The formula for continuous compounding is (A = Pe^{rt}), where (A) is the final amount, (P) is the principal amount, (r) is the annual interest rate (in decimal form), and (t) is the time in years. Given (P=$600), (A = $6000), and (r=0.05). Substitute these values into the formula: (6000=600e^{0.05t}).

Step2: Simplify the equation

Divide both sides of the equation by (600): (\frac{6000}{600}=e^{0.05t}), which simplifies to (10 = e^{0.05t}).

Step3: Take the natural logarithm of both sides

Using the property (\ln(e^{x})=x), take the natural logarithm of both sides: (\ln(10)=\ln(e^{0.05t})). Since (\ln(e^{0.05t}) = 0.05t), the equation becomes (\ln(10)=0.05t).

Step4: Solve for (t)

We know that (\ln(10)\approx2.3026). Then (t=\frac{\ln(10)}{0.05}). Substitute (\ln(10)\approx2.3026) into the formula: (t=\frac{2.3026}{0.05}=46.052).

Answer:

(t = 46.05) years (rounded to two decimal places)