Answer :
[tex]\begin{gathered} \text{Given} \\ A=15000 \\ P=6750 \\ r=5.25\%\rightarrow0.0525 \end{gathered}[/tex]
Recall the formula for the Future value of a principal amount, compounded continuously.
[tex]\begin{gathered} A=Pe^{rt} \\ \text{where} \\ A\text{ is the Future Value} \\ P\text{ is the Principal amount} \\ r\text{ is the rate in decimals} \\ t\text{ is time in years} \end{gathered}[/tex]Rearrange the equation so that we can solve for time t.
[tex]\begin{gathered} A=Pe^{rt}^{} \\ \frac{A}{P}=\frac{Pe^{rt}}{P} \\ \frac{A}{P}=e^{rt} \\ e^{rt}=\frac{A}{P} \\ \ln e^{rt}=\ln \mleft(\frac{A}{P}\mright) \\ rt\ln e^{}=\ln \mleft(\frac{A}{P}\mright) \\ rt^{}=\ln \mleft(\frac{A}{P}\mright) \\ \frac{rt}{r}=\frac{\ln (\frac{A}{P})}{r} \\ t=\frac{\ln(\frac{A}{P})}{r} \end{gathered}[/tex]Now substitute the following given and we have
[tex]\begin{gathered} t=\frac{\ln(\frac{A}{P})}{r} \\ t=\frac{\ln (\frac{15000}{6750})}{0.0525} \\ t=\frac{\ln (\frac{20}{9})}{0.0525} \\ t=15.2096704\text{ years} \end{gathered}[/tex]Rounding the answer to the nearest tenth, the investment will reach $15000, in approximately 15.2 years.
