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After the end of an advertising campaign, the sales of a product are given by

[tex]\[ S(t)=100,000 e^{-0.8t} \][/tex]

where [tex]\( S \)[/tex] is weekly sales (in dollars) and [tex]\( t \)[/tex] is the number of weeks since the end of the campaign.

(a) Find the rate of change of [tex]\( S \)[/tex] (that is, the rate of sales decay).

[tex]\[ \frac{dS}{dt} = \square \][/tex]

(b) From looking at the function and its derivative, explain how you know sales are decreasing.

The given function is a [tex]\(\square\)[/tex] model. Additionally, the derivative of the given function is always [tex]\(\square\)[/tex].


Sagot :

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.
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