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Exponential Functions

Exponential Functions 1.12 - Solution

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a

Let's determine the base of the function. y=200(0.75)t\begin{gathered} y=200({\color{#FF0000}{0.75}})^t \end{gathered} Its base is 0.75{\color{#FF0000}{0.75}} which is greater than 00 and less than 1.1. Therefore, the model represents exponential decay.

b
We will first analyze the exponential decay model. Exponential Decay Modely=a(1r)t\begin{gathered} \underline{\textbf{Exponential Decay Model}}\\ y=\textcolor{darkorange}{a}(1-\textcolor{magenta}{r})^t \end{gathered} In the model, a\textcolor{darkorange}{a} is the initial amount and r\textcolor{magenta}{r} is the percent decrease written as a decimal. Knowing that the base of the function is 0.75,0.75, we can identify the annual percent decrease.
1r=0.751-r=0.75
-r=-0.25\text{-} r=\text{-} 0.25
r=0.25r=0.25
r=25%r=25\, \%
Therefore, the annual percent decrease is 25%.25 \, \%.