Question

A big company plans to expand its franchise and, with this, its number of employees. For tax reasons it is most beneficial to expand the number of employees at a rate of 5% per year. If the company currently has 4, 730 employees, how many years will it take until the company has 6, 000 employees? Round your answer to the nearest hundredth.

Answer 1

`\bf A(t)=4730(1+r)^t\qquad \begin{cases} A(t)=\textit{amount of employees}\\ r=rate\\ t=years\\ ----------\\ A(t)=6000\\ r=5\%\to (5)/(100)\to 0.05\\ t=years \end{cases} \\\\\\ 6000=4730(1+0.05)^t\implies 6000=4730(1.05)^t\\\\\\ \cfrac{6000}{4730}=1.05^t \\\\\\ \cfrac{600}{473}=1.05^t\implies log\left( \cfrac{600}{473} \right)=log[(1.05)^t] \\\\\\ log\left( \cfrac{600}{473} \right)=t\cdot log(1.05)\implies \cfrac{log\left( (600)/(473) \right)}{log(1.05)}=t`