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Answer :
To find the current rate of change of sales, we need to understand how sales [tex]\( s \)[/tex] are affected by changes in advertising costs [tex]\( x \)[/tex].
The relationship between sales and advertising costs is given by the equation:
[tex]\[ s = 60000 - 350000 e^{-0.0003 x} \][/tex]
We are given that the current advertising cost is [tex]$2000 and that this cost is increasing at a rate of $[/tex]300 per week. We need to determine how the sales [tex]\( s \)[/tex] are changing at this point in time.
Here's a step-by-step solution:
1. Express Sales as a Function of Advertising Costs:
The sales function is:
[tex]\[ s(x) = 60000 - 350000 e^{-0.0003 x} \][/tex]
2. Find the Derivative of [tex]\( s \)[/tex] with Respect to [tex]\( x \)[/tex]:
To find the rate of change of sales with respect to advertising costs, differentiate [tex]\( s(x) \)[/tex] with respect to [tex]\( x \)[/tex]. This gives us the derivative [tex]\( \frac{ds}{dx} \)[/tex].
3. Evaluate the Derivative at the Current Advertising Cost:
Substitute [tex]\( x = 2000 \)[/tex] into the derivative to find the rate at which sales are changing with respect to advertising cost at this specific advertising cost.
The computed derivative value at [tex]\( x = 2000 \)[/tex] is approximately [tex]\( 57.63 \)[/tex].
4. Calculate the Rate of Change of Sales:
Since the advertising costs are increasing at a rate of [tex]$300 per week, the overall rate of change of sales is the product of the derivative \( \frac{ds}{dx} \) and the rate of change of advertising costs \( \frac{dx}{dt} = 300 \).
\[
\text{Rate of change of sales} = \frac{ds}{dx} \times \frac{dx}{dt} = 57.63 \times 300 \approx 17287.57
\]
Therefore, the current rate of change of sales is approximately $[/tex]17,287.57 per week.
The relationship between sales and advertising costs is given by the equation:
[tex]\[ s = 60000 - 350000 e^{-0.0003 x} \][/tex]
We are given that the current advertising cost is [tex]$2000 and that this cost is increasing at a rate of $[/tex]300 per week. We need to determine how the sales [tex]\( s \)[/tex] are changing at this point in time.
Here's a step-by-step solution:
1. Express Sales as a Function of Advertising Costs:
The sales function is:
[tex]\[ s(x) = 60000 - 350000 e^{-0.0003 x} \][/tex]
2. Find the Derivative of [tex]\( s \)[/tex] with Respect to [tex]\( x \)[/tex]:
To find the rate of change of sales with respect to advertising costs, differentiate [tex]\( s(x) \)[/tex] with respect to [tex]\( x \)[/tex]. This gives us the derivative [tex]\( \frac{ds}{dx} \)[/tex].
3. Evaluate the Derivative at the Current Advertising Cost:
Substitute [tex]\( x = 2000 \)[/tex] into the derivative to find the rate at which sales are changing with respect to advertising cost at this specific advertising cost.
The computed derivative value at [tex]\( x = 2000 \)[/tex] is approximately [tex]\( 57.63 \)[/tex].
4. Calculate the Rate of Change of Sales:
Since the advertising costs are increasing at a rate of [tex]$300 per week, the overall rate of change of sales is the product of the derivative \( \frac{ds}{dx} \) and the rate of change of advertising costs \( \frac{dx}{dt} = 300 \).
\[
\text{Rate of change of sales} = \frac{ds}{dx} \times \frac{dx}{dt} = 57.63 \times 300 \approx 17287.57
\]
Therefore, the current rate of change of sales is approximately $[/tex]17,287.57 per week.
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