Question

What is the decay factor of the exponential function represented by the table?

1/3
2/3
2
6

Answer 1

if you look at the table, we know that when x = 0, f(x) = 6, thus


`\bf \qquad \textit{Amount for Exponential Decay}\\\\ f(x)=I(1 - r)^x\qquad \begin{cases} f(x)=\textit{accumulated amount}\\ I=\textit{initial amount}\\ r=rate\to r\%\to (r)/(100)\\ x=\textit{elapsed time}\\ ----------\\ x=0\\ f(x)=6 \end{cases} \\\\\\ 6=I(1-r)^0\implies 6=I\cdot 1\implies 6=I\qquad then~~\boxed{f(x)=6(1-r)^x}`

now... let's notice from the table, when x = 1, f(x) = 2, thus


`\bf \qquad \textit{Amount for Exponential Decay}\\\\ f(x)=6(1 - r)^x\qquad \begin{cases} f(x)=\textit{accumulated amount}\\ I=\textit{initial amount}\to &6\\ r=rate\to r\%\to (r)/(100)\\ x=\textit{elapsed time}\\ ----------\\ x=1\\ f(x)=2 \end{cases} \\\\\\ 2=6(1-r)^1\implies \cfrac{2}{6}=(1-r)^1\implies \cfrac{1}{3}=1-r \\\\\\ r=1-\cfrac{1}{3}\implies r=\cfrac{2}{3}\quad\quad\quad therefore~~~~~\boxed{f(x)=6\left(1-(2)/(3) \right)^x}`