type the correct answer in the box. round your answer to the nearest whole number. andrew deposited $500 in…

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 $P = 500$, $A = 543$, and $r=0.065$. We substitute these values into the formula: $543 = 500e^{0.065t}$.

Step2: Solve for $t$

First, divide both sides of the equation by 500: $\frac{543}{500}=e^{0.065t}$, so $1.086 = e^{0.065t}$. Then, take the natural logarithm of both sides: $\ln(1.086)=\ln(e^{0.065t})$. Since $\ln(e^{x}) = x$, we have $\ln(1.086)=0.065t$. We know that $\ln(1.086)\approx0.0826$. So, $t=\frac{\ln(1.086)}{0.065}=\frac{0.0826}{0.065}\approx1.27$ years.

Step3: Convert years to months

Since there are 12 months in a year, the number of months $n = 1.27\times12 = 15.24$ months. Rounding to the nearest whole number, we get 15 months.

Answer:

15