We need the time t in the compound interest formula:
[tex]A=P(1+\frac{r}{n})^{nt}[/tex]Then, if we move P to the left hand side, we get
[tex]\frac{A}{P}=(1+\frac{r}{n})^{nt}[/tex]If we apply logarithm in both sides, we obtain
[tex]\log \frac{A}{P}=nt\cdot\log (1+\frac{r}{n})[/tex]therefore, the time is given by
[tex]t=\frac{\log \frac{A}{P}}{n\cdot\log (1+\frac{r}{n})}[/tex]In our case A= $33467.27 (Amount), P=$32000 (Principal), n=0.03 (interest rate) and n=4 (quarterly interest).
By substituting these values into the last formula, we have
[tex]t=\frac{\log \frac{33467.27}{32000}}{4\cdot\log (1+\frac{0.03}{4})}[/tex]which gives
[tex]t=\frac{\log 1.04585}{4\cdot\log 1.0075}[/tex]the, the time is
[tex]\begin{gathered} t=\frac{0.04482}{0.02988} \\ t=1.5 \end{gathered}[/tex]that is, the time will be equal to 1.5 years.
