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 ddaeeric 2020-12-02 Answered 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
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Analyse the given information Since the interest is compounded continuously, the Principal will increase exponentially. The amount after t years is given by $A\left(t\right)=P{e}^{rt}$ It is given that the initial investment,$P=\mathrm{}2000$ and the interest rate per year,$r=4.5\mathrm{%}=0.045$ 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 $A\left(t\right)=P{e}^{rt}$ After two years: t=2 $A\left(2\right)=2000{e}^{0.045\left(2\right)}$
$A\left(2\right)=2000{e}^{0.09}$
$A\left(2\right)=\mathrm{}2188.35$ [rounded to two decimals(nearest cent) ] Amount after 2 years is $2188.35 b) After two years i.e t=4 $A\left(t\right)=P{e}^{rt}$ After two years: t=4 $A\left(4\right)=2000{e}^{0.045\left(4\right)}$ $A\left(4\right)=2000{e}^{0.18}$ $A\left(4\right)=\mathrm{}2394.43$ [rounded to two decimals(nearest cent) ] Amount after 4 years is$2394.43 c) After twelve years i.e t=12 $A\left(t\right)=P{e}^{rt}$ After two years: t=12 $A\left(12\right)=2000{e}^{0.045\left(12\right)}$
$A\left(12\right)=2000{e}^{0.54}$
$A\left(12\right)=\mathrm{}3432.01$ [rounded to two decimals(nearest cent)] Amount after 12 years is \$3432.01