The Solution:
Given:
[tex]\begin{gathered} Formula:\text{ }A=P(1+i)^n \\ \text{ In this case,} \\ A=\text{ \$}22000 \\ P=\text{ \$}2000 \\ n=4\times t=4t\text{ \lparen t=number of years\rparen} \\ i=\frac{8.2}{100\times4}=\frac{8.2}{400}=0.0205 \end{gathered}[/tex]Substitute these values in the formula.
[tex]\begin{gathered} 22000=2000(1+0.0205)^{4t} \\ \\ Dividing\text{ both sides by 2000, we get} \\ \\ (1.0205)^{4t}=\frac{22000}{2000} \\ \\ (1.0205)^{4t}=11 \end{gathered}[/tex]Taking the ln of both sides, we get:
[tex]\begin{gathered} \ln(1.0205)^{4t}=\ln(11) \\ \\ 4t\ln(1.0205)=\ln(11) \\ \\ 4t=\frac{\ln(11)}{\ln(1.0205)} \end{gathered}[/tex][tex]t=\frac{\ln(11)}{4\ln(1.0205)}=29.54134\approx30\text{ years}[/tex]Therefore, the correct answer is 30 years.
