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

Explanation:

Step1: Recall 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 $P = 600$, $r=0.05$, and $A = 6000$. Substitute these values into the formula: $6000=600e^{0.05t}$.

Step2: Simplify the equation

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

Step3: Take the natural logarithm of both sides

Since $\ln(e^{x})=x$, taking the natural logarithm of both sides of the equation $10 = e^{0.05t}$ gives $\ln(10)=\ln(e^{0.05t})$. So, $\ln(10)=0.05t$.

Step4: Solve for $t$

We know that $\ln(10)\approx2.3026$. Then, $t=\frac{\ln(10)}{0.05}$. Substitute the value of $\ln(10)$: $t=\frac{2.3026}{0.05}=46.052$.

Answer:

$46.052$