If $2000 is invested at an interest rate of 4.5% per year, compounded continuously, find the value of the investment after the given number of years. (Round your answers to the nearest cent.) a) 2 years b) 4 years c) 12 years

Question
Decimals
asked 2020-12-02
If $2000 is invested at an interest rate of 4.5% per year, compounded continuously, find the value of the investment after the given number of years. (Round your answers to the nearest cent.) a) 2 years b) 4 years c) 12 years

Answers (1)

2020-12-03
Analyse the given information Since the interest is compounded continuously, the Principal will increase exponentially. The amount after t years is given by \(\displaystyle{A}{\left({t}\right)}={P}{e}^{{{r}{t}}}\) It is given that the initial investment,\(\displaystyle{P}=\${2000}\) and the interest rate per year,\(\displaystyle{r}={4.5}\%={0.045}\) Process: In each part a), b) and c) plug in values of P, r and t in each case to find the value of Amount after a given number of years. To round to nearest cent we need to round answers to two decimals. a) After two years i.e t=2 \(\displaystyle{A}{\left({t}\right)}={P}{e}^{{{r}{t}}}\) After two years: t=2 \(\displaystyle{A}{\left({2}\right)}={2000}{e}^{{{0.045}{\left({2}\right)}}}\)
\(\displaystyle{A}{\left({2}\right)}={2000}{e}^{{{0.09}}}\)
\(\displaystyle{A}{\left({2}\right)}=\${2188.35}\) [rounded to two decimals(nearest cent) ] Amount after 2 years is $2188.35 b) After two years i.e t=4 \(\displaystyle{A}{\left({t}\right)}={P}{e}^{{{r}{t}}}\) After two years: t=4 \(\displaystyle{A}{\left({4}\right)}={2000}{e}^{{{0.045}{\left({4}\right)}}}\)
\(\displaystyle{A}{\left({4}\right)}={2000}{e}^{{{0.18}}}\)
\(\displaystyle{A}{\left({4}\right)}=\${2394.43}\) [rounded to two decimals(nearest cent) ] Amount after 4 years is $2394.43 c) After twelve years i.e t=12 \(\displaystyle{A}{\left({t}\right)}={P}{e}^{{{r}{t}}}\) After two years: t=12 \(\displaystyle{A}{\left({12}\right)}={2000}{e}^{{{0.045}{\left({12}\right)}}}\)
\(\displaystyle{A}{\left({12}\right)}={2000}{e}^{{{0.54}}}\)
\(\displaystyle{A}{\left({12}\right)}=\${3432.01}\) [rounded to two decimals(nearest cent)] Amount after 12 years is $3432.01
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