Question

Every two years, a manufacturer's total yearly sales decline 20%. She sells 400 total units in her first year. Rounded to the nearest unit, how many total units will she sell her ninth year?

Answer 1

$\bf \textit{Periodic Exponential Decay}\\\\ A=I(1 - r)^{\frac{t}{p}}\qquad \begin{cases} A=\textit{accumulated amount}\\ I=\textit{initial amount}\\ r=rate\to r\%\to \frac{r}{100}\\ t=\textit{elapsed time}\\ p=period \end{cases}$

we know, the first year she sold 400 units, thus year 0, 400 units, t = 0, A = 400.

$\bf \textit{Periodic Exponential Decay}\\\\ A=I(1 - r)^{\frac{t}{p}}\qquad \begin{cases} A=\textit{accumulated amount}\to &400\\ I=\textit{initial amount}\\ r=rate\to r\%\to \frac{r}{100}\\ t=\textit{elapsed time}\to &0\\ p=period \end{cases} \\\\\\ 400=I(1-r)^{\frac{0}{p}}\implies 400=I\cdot 1\implies 400=I$

therefore then

$\bf \textit{Periodic Exponential Decay}\\\\ A=I(1 - r)^{\frac{t}{p}}\qquad \begin{cases} A=\textit{accumulated amount}\\ I=\textit{initial amount}\to &400\\ r=rate\to 20\%\to \frac{20}{100}\to &0.20\\ t=\textit{elapsed time}\\ p=period\to &2 \end{cases} \\\\\\ A=400(1-0.2)^{\frac{t}{2}}$

now, how many units will it had decreased by the 9th year?  t = 9

$\bf A=400(1-0.2)^{\frac{9}{2}}$

Answer 2

400*(1-0.10)^9=

400*0.9^9 =

400* 0.387420489=

154.968

 round to 155