Question

The sales decay for a product is given by S = 70000e^ -0.8x where S is the monthly sales and x is the number of months that have passed since the end of a promotional campaign. How many months after the end of the campaign will sales drop below 1000, if no new campaign is initiated? (Round your answers to two decimal places.) What will be the sales 5 months after the end of the campaign?

Answer 1

Answer:

After 5.31 months sales will drop below 1000 and 5 months after the end of the campaign sales will be 1282.09

Step-by-step explanation:

Let's find the solutions for the two questions.

First question: How many months after the end of the campaign will sales drop below 1000.

Because the problem asks for how many months, and since 'x' represents month variable, then the problem is asking for 'x'.

Using the same equation for sales we can observe the following:

$S=70000*e^{-0.8X}$, but we have S which is 1000, so:

$1000=70000*e^{-0.8X}$ which is equal to:

$1000/70000=e^{-0.8X}$ which is equal to:

$1/70=e^{-0.8X}$ by applying ln(x) properties:

$ln(1/70)=ln(e^{-0.8X})$ which is equal to:

$ln(1/70)=-0.8X$ which is equal to:

$ln(1/70)/(-0.8)=X$ so:

$X=5.31 months$

Second question: what will be the sales 5 months after the end of the campaign.

Because the problem asks for what will be the sales, and since 'S' represents the sales, then the problem is asking for 'S'.

Using the same equation for sales we can observe the following:

$S=70000*e^{-0.8X}$, but we have x which is 5 months, so:

$S=70000*e^{-0.8*5}$ which is equal to

$S=1282.09$

In conclusion, after 5.31 months sales will drop below 1000 and 5 months after the end of the campaign sales will be 1282.09.