Answer:
The current rate of change of sales is ​$4 per week.
Step-by-step explanation:
Let as consider the weekly sales function
[tex]s=40,000-30,000e^{-0.0004x}[/tex]
where, s represents the weekly sales and x is the weekly advertising costs (both in​ dollars).
Differentiate with respect to x.
[tex]\dfrac{ds}{dx}=\dfrac{d}{dx}(40,000-30,000e^{-0.0004x})[/tex]
[tex]\dfrac{ds}{dx}=\dfrac{d}{dx}(40,000)-30,000\dfrac{d}{dx}(e^{-0.0004x})[/tex]
[tex]\dfrac{ds}{dx}=0-30,000(-0.0004e^{-0.0004x})[/tex]
[tex]\dfrac{ds}{dx}=12e^{-0.0004x}[/tex] Â Â Â Â Â Â .... (1)
Here, [tex]\dfrac{ds}{dx}[/tex] represents the rate of change of sales at weekly advertising costs x.
Current weekly advertising costs = $2500
Substitute x=2500 in equation (1).
[tex]\dfrac{ds}{dx}_{x=2500}=12e^{-0.0004(2500)}[/tex]
[tex]\dfrac{ds}{dx}_{x=2500}=4.4145533[/tex]
[tex]\dfrac{ds}{dx}_{x=2500}=4[/tex]
Therefore, the current rate of change of sales is ​$4 per week.
The current rate of change is $4.41 per week
The weekly sales is given as:
[tex]s = 40000 - 30000e^{-0.0004x}[/tex]
Differentiate the above equation to determine the rate of change
[tex]s' = 0 +0.0004 * 30000e^{-0.0004x}[/tex]
Evaluate the product
[tex]s' = 0 +12e^{-0.0004x}[/tex]
Evaluate the sum
[tex]s' = 12e^{-0.0004x}[/tex]
The current weekly advertisement cost is $2500.
So, we have:
[tex]s' = 12e^{-0.0004*2500}[/tex]
Evaluate the product
[tex]s' = 12e^{-1}[/tex]
Evaluate the expression
[tex]s' = 4.41[/tex]
Hence, the current rate of change is $4.41 per week
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