We have that the formula for the compound interest semi annually is:
[tex]A=P(1+\frac{r}{2})^{2t}[/tex]where A is the total amount, P is the principal amount, r is the interest rate and t is the time in years.
In this case, we have the following information:
[tex]\begin{gathered} r=6\%=0.06 \\ A=40000+13755.66=53755.66 \\ P=40000 \end{gathered}[/tex]then, using the first equation, we get:
[tex]\begin{gathered} 53755.66=40000(1+\frac{0.06}{2})^{2t} \\ \Rightarrow\frac{53755.66}{40000}=(1+0.03)^{2t} \\ \Rightarrow1.3439=(1.03)^{2t} \end{gathered}[/tex]using logarithm on both sides of the equation, we get the following:
[tex]\begin{gathered} \ln (1.3439)=\ln (1.03^{2t}) \\ \Rightarrow\ln (1.3439)=2t\ln (1.03) \\ \Rightarrow t=\frac{\ln (1.3439)}{2\ln (1.03)}=4.99\approx5 \\ t\approx5 \end{gathered}[/tex]therefore, it wil take approximately 5 years to get the desired interest.
