Question

When Tobey was born, his parents invested $2,000 in a fund that paid an

annual interest of 6%. How old will Tobey be when the investment is worth
at least $5,000?

Answer 1

`~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+(r)/(n)\right)^(nt) \quad \begin{cases} A=\textit{accumulated amount}\dotfill & \$ 5000\\ P=\textit{original amount deposited}\dotfill &\$2000\\ r=rate\to 6\%\to (6)/(100)\dotfill &0.06\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{annually, thus once} \end{array}\dotfill &1\\ t=years \end{cases}`

`5000 = 2000\left(1+(0.06)/(1)\right)^(1\cdot t) \implies \cfrac{5000}{2000}=(1+0.06)^t\implies \cfrac{5}{2}=1.06^t \\\\\\ \log\left( \cfrac{5}{2} \right)=\log(1.06^t)\implies \log\left( \cfrac{5}{2} \right)=t\log(1.06) \\\\\\ \cfrac{ ~~ \log\left( (5)/(2) \right) ~~ }{\log(1.06)}=t\implies 15.73\approx t\qquad \textit{15 years and about 9 months}`