Investment: $1600
Nominal rate of interest: 7% = 0.07
Composition: Quarterly (n = 4)
Final growth: $7500
Then, using the formula:
[tex]7500=1600(1+\frac{0.07}{4})^{4t}[/tex]Now, solving this equation for t:
[tex]\begin{gathered} 4.6875=1.0175^{4t} \\ 4.6875=1.0175^{4t} \\ \ln (4.6875)=\ln (1.0175^{4t}) \\ \ln (4.6875)=4t\ln (1.0175^{}) \\ t=\frac{\ln (4.6875)}{4\cdot\ln (1.0175^{})} \\ \therefore t=22.2625\text{ years} \end{gathered}[/tex]