College

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

[tex]s = 60000 - 320000 e^{-0.0009 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 :

We start with the relationship between the weekly sales [tex]\( s \)[/tex] and the weekly advertising costs [tex]\( x \)[/tex] given by

[tex]$$
s=60000-320000e^{-0.0009x}.
$$[/tex]

We are told that the advertising costs are currently [tex]\( x=2000 \)[/tex] dollars and that the advertising costs are increasing at a rate of

[tex]$$
\frac{dx}{dt}=300 \text{ dollars per week}.
$$[/tex]

Our goal is to find the current rate of change of sales [tex]\( \frac{ds}{dt} \)[/tex], which we will obtain by using the chain rule:

[tex]$$
\frac{ds}{dt}=\frac{ds}{dx}\cdot\frac{dx}{dt}.
$$[/tex]

### Step 1. Find [tex]\(\frac{ds}{dx}\)[/tex]

Differentiate [tex]\( s \)[/tex] with respect to [tex]\( x \)[/tex]:

[tex]$$
\frac{ds}{dx}=\frac{d}{dx}\left[60000-320000e^{-0.0009x}\right].
$$[/tex]

The derivative of the constant [tex]\( 60000 \)[/tex] is [tex]\( 0 \)[/tex]. For the second term, apply the chain rule. The derivative of [tex]\( e^{-0.0009x} \)[/tex] is

[tex]$$
\frac{d}{dx}\left(e^{-0.0009x}\right)=-0.0009e^{-0.0009x}.
$$[/tex]

Thus,

[tex]$$
\frac{ds}{dx}=-320000\left(-0.0009e^{-0.0009x}\right)=320000\cdot0.0009e^{-0.0009x}.
$$[/tex]

Note that

[tex]$$
320000\cdot0.0009=288,
$$[/tex]

so we can write

[tex]$$
\frac{ds}{dx}=288e^{-0.0009x}.
$$[/tex]

### Step 2. Evaluate [tex]\(\frac{ds}{dx}\)[/tex] at [tex]\( x=2000 \)[/tex]

Substitute [tex]\( x=2000 \)[/tex] into the expression:

[tex]$$
\frac{ds}{dx}=288e^{-0.0009(2000)}.
$$[/tex]

Calculate the exponent:

[tex]$$
0.0009\times2000=1.8,
$$[/tex]

which gives

[tex]$$
\frac{ds}{dx}=288e^{-1.8}.
$$[/tex]

Numerically, [tex]\( e^{-1.8}\approx0.1652988882 \)[/tex]. Therefore,

[tex]$$
\frac{ds}{dx}\approx288\times0.1652988882\approx47.60608.
$$[/tex]

### Step 3. Compute [tex]\(\frac{ds}{dt}\)[/tex]

Now, use the chain rule:

[tex]$$
\frac{ds}{dt}=\frac{ds}{dx}\cdot\frac{dx}{dt}.
$$[/tex]

Substitute the computed values:

[tex]$$
\frac{ds}{dt}\approx47.60608\times300.
$$[/tex]

Multiplying these,

[tex]$$
\frac{ds}{dt}\approx14281.82394.
$$[/tex]

### Final Answer

The current rate of change of sales is approximately

[tex]$$
\boxed{14281.82 \text{ dollars per week}}.
$$[/tex]