select the correct answer. mary deposited $350 in a bank account that promises 2.8 percent interest…

select the correct answer. mary deposited $350 in a bank account that promises 2.8 percent interest compounded continuously. approximately how many years will it take to reach a balance of $500?\n\na. 1.43 years\nb. 2.80 years\nc. 5.55 years\nd. 12.77 years

select the correct answer. mary deposited $350 in a bank account that promises 2.8 percent interest compounded continuously. approximately how many years will it take to reach a balance of $500?\n\na. 1.43 years\nb. 2.80 years\nc. 5.55 years\nd. 12.77 years

Answer

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 $P = 350$, $A = 500$, and $r=0.028$. Substitute these values into the formula: $500 = 350e^{0.028t}$.

Step2: Solve for $e^{0.028t}$

Divide both sides of the equation by 350: $\frac{500}{350}=e^{0.028t}$, which simplifies to $\frac{10}{7}=e^{0.028t}$.

Step3: Take the natural logarithm of both sides

$\ln(\frac{10}{7})=\ln(e^{0.028t})$. Since $\ln(e^{x}) = x$, the right - hand side simplifies to $0.028t$. And $\ln(\frac{10}{7})\approx\ln(1.4286)\approx0.3567$. So, $0.3567 = 0.028t$.

Step4: Solve for $t$

Divide both sides by 0.028: $t=\frac{0.3567}{0.028}\approx12.74$.

Answer:

D. 12.77 years