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

\(\displaystyle{A}{\left({2}\right)}={2000}{e}^{{{0.09}}}\)

\(\displaystyle{A}{\left({2}\right)}=\${2188.35}\) [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