College

A retail store estimates that weekly sales [tex]s[/tex] and weekly advertising costs [tex]x[/tex] (both in dollars) are related by the equation:

[tex]s = 60000 - 350000 e^{-0.0003 x}[/tex]

The current weekly advertising costs are 2000 dollars, and these costs are increasing at the rate of 300 dollars per week. Find the current rate of change of sales.

Rate of change of sales [tex]= \square[/tex]

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.