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Question

# When interest is compounded continuously, the amount of money increases at a rate proportional to the amount S present at time t, that is, d S / d t=r S, where r is the annual rate of interest.Find the amount of money accrued at the end of 5 years when $5000 is deposited in a savings account drawing$5 $\frac{3}{4}$ %\$ annual interest compounded continuously.

Solution

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We have the amount of money as $S$ and this amount of money $S$ increases at time $t$ at a rate is proportional to $S$ as

$\begin{gather*} \dfrac{dS}{dt} = r \ S \tag{1} \end{gather*}$

where $r$ is the annual rate of interest, with the condition

$\begin{gather*} S (t = 0) = 5,000 \ \text{dollars} \end{gather*}$

with an annual interest as $r = \dfrac{5 \frac{3}{4}}{100} = \dfrac{\frac{23}{4}}{100} = 0.0575$, and we have to obtain the amount of accrued money after $5 \ \text{years}$.

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