Answer
13.9 years
Work Step by Step
After $t$ years, the balance, $A$, in an account with principal $P$ and annual
interest rate $r$ (in decimal form) is given by one of the following formulas:
1. For $n$ compoundings per year: $A=P(1+\displaystyle \frac{r}{n})^{nt}$
2. For continuous compounding: $A=Pe^{rt}$.
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We insert the given information,
n=4, P=4000, A=8000, r=0.005
into formula (1), and solve for t:
$8000=4000(1+\displaystyle \frac{0.05}{4})^{4t}\qquad/\div 4000$
$ 2=1.0125^{4t}\qquad$... apply ln( ) to both sides...
$\ln 2=4t\cdot\ln 1.0125\qquad /\div(4\cdot\ln 1.0125)$
$t=\displaystyle \frac{\ln 2}{4\cdot\ln 1.0125}\approx 13.9$ years
