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A retail store estimates that weekly sales \( s \) and weekly advertising costs \( x \) (both in dollars) are related by

\[ s = 60000 - 350000 e^{-0.0004x} \]

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 =

Answer :

The current rate of change of sales is approximately 157.15 dollars per week.

To find the current rate of change of sales (ds/dx) with respect to weekly advertising costs (x), we need to take the derivative of the sales function (s) with respect to x.

Given the sales function:

s = 60000 − 350000[tex]e^{(-0.0004X)}[/tex]

Let's differentiate both sides of the equation with respect to x:

ds/dx = d/dx (60000 − 350000[tex]e^{(-0.0004X)}[/tex]

To differentiate the function, we need to apply the chain rule. The chain rule states that if we have a function g(f(x)), then its derivative with respect to x is given by g'(f(x) × f'(x).

In our case, g(u) = 60000 - 350000[tex]e^{u}[/tex], and f(x) = -0.0004x.

Now, let's find the derivatives:

g'(u) = d/du (60000 - 350000[tex]e^{u}[/tex]) = -350000e^u

f'(x) = d/dx (-0.0004x) = -0.0004

Using the chain rule, we get;

ds/dx = g'(f(x) × f'(x)

ds/dx = (-350000[tex]e^{(-0.0004X)}[/tex] × (-0.0004)

Now, we need to find the current rate of change of sales (ds/dx) when the current weekly advertising costs are $2000 (x = 2000). Additionally, the advertising costs are increasing at the rate of $300 per week, which means dx/dt = 300 (where t represents the number of weeks).

Now, we can calculate the current rate of change of sales (ds/dx) at x = 2000:

ds/dx = (-350000[tex]e^{(-0.0004X)}[/tex] × 2000) × (-0.0004)

ds/dx = (-350000[tex]e^{(-0.8)}[/tex] × (-0.0004)

To get the numerical value, let's evaluate [tex]e^{(-0.8)}[/tex]:

[tex]e^{(-0.8)}[/tex] ≈ 0.449

Now, substitute this value back into the equation:

ds/dx ≈ (-350000 × 0.449) × (-0.0004)

ds/dx ≈ 157.15

Therefore, the current rate of change of sales is approximately 157.15 dollars per week.

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