Question

Please hell witth these questions

4) How long will it take a \( \$ 4,000 \) investment to grow to \( \$ 7,000 \) if it earns \( 5 \% \) compounded semiannually? Show " \( N \) " with two places after the decimal. Round your final answ

Answer 1

`~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+(r)/(n)\right)^(nt) \quad \begin{cases} A=\textit{accumulated amount}\dotfill & \$ 7000\\ P=\textit{original amount deposited}\dotfill &\$4000\\ r=rate\to 5\%\to (5)/(100)\dotfill &0.05\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{semi-annually, thus two} \end{array}\dotfill &2\\ t=years \end{cases}`

`7000 = 4000\left(1+(0.05)/(2)\right)^(2\cdot t) \implies \cfrac{7000}{4000}=1.025^(2t)\implies \cfrac{7}{4}=1.025^(2t) \\\\\\ \log\left( \cfrac{7}{4} \right)=\log\left( 1.025^(2t) \right)\implies \log\left( \cfrac{7}{4} \right)=t\log\left( 1.025^(2) \right) \\\\\\ \cfrac{ ~~ \log\left( (7)/(4) \right) ~~ }{\log\left( 1.025^(2) \right)}=t\implies 11.33\approx t\qquad \textit{about 11 years and 4 months}`