To answer this question we will use the following formula for compounded daily interest:
[tex]\begin{gathered} A=A_0(1+\frac{r}{365})^{365t},_{} \\ \text{where A}_0\text{ is the initial amount, r is the annual rate as a decimal number, } \\ \text{and t is the number of years.} \end{gathered}[/tex]Substituting A₀=8700, A=12000, and r=0.02 we get:
[tex]12000=8700(1+\frac{0.02}{365})^{365t}\text{.}[/tex]Dividing the above result by 8700 we get:
[tex]\begin{gathered} \frac{12000}{8700}=\frac{8700}{8700}(1+\frac{0.02}{365})^{365t}, \\ \frac{40}{29}=(1+\frac{0.02}{365})^{365t}\text{.} \end{gathered}[/tex]Applying the natural logarithm we get:
[tex]\ln (\frac{40}{29})=365t\cdot\ln (1+\frac{0.02}{365})\text{.}[/tex]Finally, dividing the above equation by
[tex]365\cdot\ln (1+\frac{0.02}{365})[/tex]we get:
[tex]t=\frac{\ln(\frac{40}{29})}{365\ln(1+\frac{0.02}{365})}\text{.}[/tex]Therefore:
[tex]t\approx16.08[/tex]Answer: The value of the fund reaches $12,000 after 16.08 years.
