type the correct answer in the box. round your answer to the hundredth. an investment in a savings account…

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 number of years. We know that $A = 3P$ and $r=0.05$. Substituting these values into the formula gives $3P=Pe^{0.05t}$.

Step2: Simplify the equation

Divide both sides of the equation $3P = Pe^{0.05t}$ by $P$ (since $P\neq0$). We get $3 = e^{0.05t}$.

Step3: Take the natural logarithm of both sides

Taking the natural logarithm of both sides, $\ln(3)=\ln(e^{0.05t})$. Since $\ln(e^{x}) = x$, the right - hand side simplifies to $0.05t$. So, $\ln(3)=0.05t$.

Step4: Solve for $t$

We can solve for $t$ by dividing both sides of the equation $\ln(3)=0.05t$ by $0.05$. So, $t=\frac{\ln(3)}{0.05}$. We know that $\ln(3)\approx1.0986$. Then $t=\frac{1.0986}{0.05}=21.972$.

Step5: Round the answer

Rounding $21.972$ to the hundredth, we get $t = 21.97$.

Answer:

$21.97$