To address the given questions step-by-step:
### Part (a)
We need to find the rate of change of [tex]\( S(t) \)[/tex], which is represented by the derivative of [tex]\( S(t) \)[/tex] with respect to [tex]\( t \)[/tex].
Given the function:
[tex]\[
S(t) = 100,000 e^{-0.8 t}
\][/tex]
1. To find the derivative [tex]\(\frac{dS}{dt}\)[/tex], we must differentiate [tex]\( S(t) \)[/tex] with respect to [tex]\( t \)[/tex].
2. Using the chain rule, the differentiation of an exponential function [tex]\( a e^{kt} \)[/tex] with respect to [tex]\( t \)[/tex] is [tex]\( a k e^{kt} \)[/tex].
Thus,
[tex]\[
\frac{dS}{dt} = 100,000 \cdot \frac{d}{dt}\left( e^{-0.8 t} \right)
\][/tex]
3. Applying the chain rule to [tex]\( e^{-0.8 t} \)[/tex], we get:
[tex]\[
\frac{d}{dt}\left( e^{-0.8 t} \right) = -0.8 e^{-0.8 t}
\][/tex]
Therefore,
[tex]\[
\frac{dS}{dt} = 100,000 \cdot (-0.8) e^{-0.8 t}
\][/tex]
[tex]\[
\frac{dS}{dt} = -80,000 e^{-0.8 t}
\][/tex]
Hence, the rate of change of [tex]\( S \)[/tex] is:
[tex]\[
\frac{dS}{dt} = -80,000 e^{-0.8 t}
\][/tex]
### Part (b)
To determine whether sales are decreasing, we should analyze both the function [tex]\( S(t) \)[/tex] and its derivative [tex]\(\frac{dS}{dt}\)[/tex].
1. The given function [tex]\( S(t) = 100,000 e^{-0.8 t} \)[/tex] is an exponential decay model. This is because it involves an exponential function with a negative exponent, representing a decreasing trend over time.
2. The derivative [tex]\(\frac{dS}{dt} = -80,000 e^{-0.8 t}\)[/tex] is our key indicator. Notice the following about the derivative:
- The term [tex]\( e^{-0.8 t} \)[/tex] is always positive for any value of [tex]\( t \geq 0 \)[/tex].
- The coefficient [tex]\(-80,000\)[/tex] makes [tex]\(\frac{dS}{dt}\)[/tex] always negative.
Since the derivative [tex]\(\frac{dS}{dt}\)[/tex] is always negative, it indicates that [tex]\( S(t) \)[/tex] is a decreasing function. This tells us that sales are continuously dropping over time.
Thus, the correct statements are:
- The given function is an exponential decay model.
- Additionally, the derivative of the given function is always negative, indicating sales are decreasing.
