Solution:
Given:
[tex]\begin{gathered} Pr\text{ incipal, P=\$5000} \\ \text{Amount, A=\$8500} \\ \text{Rate, r=7.5\%=}\frac{7.5}{100}=0.075 \\ \text{Number of compounding (quarterly), n =4} \\ t=\text{?} \end{gathered}[/tex]
The formula for finding the amount in compound interest is given by;
[tex]A=P(1+\frac{r}{n})^{nt}[/tex]
Substituting the values to get the time it will take,
[tex]\begin{gathered} A=P(1+\frac{r}{n})^{nt} \\ 8500=5000(1+\frac{0.075}{4})^{4t} \\ \frac{8500}{5000}=(1+0.01875)^{4t} \\ 1.7=1.01875^{4t} \\ \text{Taking the logarithm of both side to get t,} \end{gathered}[/tex][tex]\begin{gathered} 1.7=1.01875^{4t} \\ \log 1.7=\log 1.01875^{4t} \\ \text{Applying the law of logarithm below;} \\ \log a^x=x\log a \\ \text{Then,} \\ \log 1.7=4t\log 1.01875 \\ \text{Dividing both sides by log1.01875} \\ \frac{\log 1.7}{\log 1.01875}=4t \\ \frac{0.2304}{0.008068}=4t \\ 28.557=4t \\ \text{Dividing both sides by 4 to get the value of t,} \\ t=\frac{28.557}{4} \\ t=7.13925 \\ t\approx7.14years \end{gathered}[/tex]
Therefore, the time is approximately 7.14years.
