Question

Between 2000 and 2010, ACME Widgets sold widgets at a continuous rate of $R=R_{0} e^{0.125 t}$ widgets per year, where t is time in years since January 1, 2000. Suppose they were selling widgets at a rate of 1000 per year on January 1, 2000. How many widgets did they sell between 2000 and 2010? How many did they sell if the rate on January 1, 2000 was 1,000,000 widgets per year?

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$R(t)=R_0\,e^{0.125t} \quad \& \quad R_0=1000$

$\Rightarrow \quad \text {Rate of selling: } \quad \boxed {R(t)=1000\,e^{0.125t} }$

The total quantity of widgets sold between $t=0$ (Year 2000.) and $t=10$ (Year 2010.) is given by

\begin{align*} \int_{0}^{10} 1000\,e^{0,125t}\,dt=\frac {1000} {0.125}\,e^{0.125t}\,\bigg|_{0}^{10}=8000\,(e^ {1.25 }-1) \approx \color{#19804f} \boxed {\color {black} 19923 } \end{align*}

If the rate on January 1, 2000. was $10^6$ widgets per year $\quad \Rightarrow \quad R_0=10^6$

$\Rightarrow \quad \text {Rate of selling: } \quad R(t)=10^6\,e^{0.125t}$

The total quantity is

\begin{align*} \int_{0}^{10} 10^6\,e^{0,125t}\,dt=\frac {10^6} {0.125}\,e^{0.125t}\,\bigg|_{0}^{10}=(8\cdot 10^6)\cdot(e^ {1.25}-1) \approx \color{#4257b2} \boxed {\color {black} 19922744 } \end{align*}

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