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# A city's population is 1000 and growing at $5\%$ a year.Find a formula for the population at time $t$ years from now assuming that the $5\%$ per year is an: (i) Annual rate (ii) Continuous annual rate

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#### $\pmb{(i)}$

To find the formula for the population at time $t$ years from now we will use the information that is given in the text of the exercise. We now that:

\begin{align*} P(0)&=P_0=1000\tag{\footnotesize\textcolor{#4257b2}{initial population}}\\ r&=5\%=0.05\tag{\footnotesize\textcolor{#4257b2}{annual growth rate}}\\ & \Rightarrow a=1+r=1.05 \end{align*}

We see that the formula for the city's population in this case is:

$\boxed{P(t)=1000\cdot (1.05)^t}$

#### $\pmb{(ii)}$

In this case (continuous annual rate) we have that:

\begin{align*} P_0&=1000\\ k&=5\%=0.05 \end{align*}

so we have that the formula for the city's population is given by:

$\boxed{ P(t)=1000\cdot e^{0.05\cdot t} }$

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