Question

Use the formula for continuous compounding to compute the balance in the following accounts after 11, 55, and 2020 years. Also, find the APY\mathrm{APY} for each account.

A $7000\$ 7000 deposit in an account with an APR\mathrm{APR} of 4.5%4.5 \%

Solution

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Answered 2 years ago
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In the given situation, we need to determine the accumulated values of an investment of $7,000\textcolor{#c34632}{\$7,000} at an interest rate of 4.5%\textcolor{#c34632}{4.5\%} compounded continuously after 1\textcolor{#c34632}{1}, 5\textcolor{#c34632}{5}, and 20\textcolor{#c34632}{20} years.


We will use the compound interest formula for continuous compounding given by:

A=P×e(APR×Y)A = P \times e^{(APR \times Y)}

where AA is the accumulated balance after YY years, PP is the starting principal, APRAPR is the annual percentage rate in decimal form, and YY is the time in years.


Plugging in the known values, we have the following:

After 1 year:

A1=$7,000×e(0.045×1)$7,322.20\begin{aligned} A_1 &= \$7,000 \times e^{(0.045 \times 1)} \\ &\approx \textcolor{#4257b2}{\$7,322.20} \end{aligned}

After 5 years:

A5=$7,000×e(0.045×5)$8,766.26\begin{aligned} A_5&= \$7,000 \times e^{(0.045 \times 5)} \\ &\approx \textcolor{#4257b2}{\$8,766.26} \end{aligned}

After 20 years:

A20=$7,000×e(0.045×20)$17,217.22\begin{aligned} A_{20} &= \$7,000 \times e^{(0.045 \times 20)} \\ &\approx \textcolor{#4257b2}{\$17,217.22} \end{aligned}

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