alexa invested $54,000 in an account paying an interest rate of 4% compounded continuously. assuming no…

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 $P = 54000$, $r=0.04$, and $A = 106200$. Substitute these values into the formula: $106200=54000e^{0.04t}$.

Step2: Isolate the exponential term

Divide both sides of the equation by $54000$: $\frac{106200}{54000}=e^{0.04t}$, which simplifies to $\frac{59}{30}=e^{0.04t}$.

Step3: Take the natural logarithm of both sides

$\ln(\frac{59}{30})=\ln(e^{0.04t})$. Since $\ln(e^{x}) = x$, the right - hand side simplifies to $0.04t$. So, $\ln(\frac{59}{30}) = 0.04t$.

Step4: Solve for $t$

First, calculate $\ln(\frac{59}{30})\approx\ln(1.9667)\approx0.675$. Then, $t=\frac{\ln(\frac{59}{30})}{0.04}=\frac{0.675}{0.04}=16.875$.

Answer:

$17$