$500 were deposited into an account with a 8.5% interest rate, compounded continuously. how many years was…

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 $P = 500$, $r=0.085$, and $A = 3500$. Substitute these values into the formula: $3500=500e^{0.085t}$.

Step2: Solve for $e^{0.085t}$

Divide both sides of the equation $3500 = 500e^{0.085t}$ by $500$. $\frac{3500}{500}=e^{0.085t}$, so $7 = e^{0.085t}$.

Step3: Take the natural logarithm of both sides

Since $\ln(e^{x})=x$, take the natural logarithm of both sides of the equation $7 = e^{0.085t}$. $\ln(7)=\ln(e^{0.085t})$, which simplifies to $\ln(7)=0.085t$.

Step4: Solve for $t$

We know that $\ln(7)\approx1.94591$, so $t=\frac{\ln(7)}{0.085}$. $t=\frac{1.94591}{0.085}\approx22.9$.

Answer:

$t\approx22.9$ years