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Question

# Sales of a product, under relatively stable market conditions but in the absence of promotional activities such as advertising, tend to decline at a constant yearly rate. This rate of sales decline varies considerably from product to product, but it seems to remain the same for any particular product. The sales decline can be expressed by the function $S(t)=S_{0} e^{-a t}$, where S(t) is the rate of sales at time t measured in years, $S_0$ is the rate of sales at time t=0, and a is the sales decay constant. (a) Suppose the sales decay constant for a particular product is a=0.10. Let $S_0$=50.000 and find S(1) and S(3) to the nearest thousand. (b) Find S(2) and S(10) to the nearest thousand if $S_0$=80.000 and a=0.05.

Solution

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Given the function for the sales decline:

$S(t)=S_{0}e^{-at}$

where $S(t)$ is the rate of sales at time $t$ measured in years, $S_{0}$ is the rate of sales at time $t=0$, and $a$ is the sales delay constant.

For problem $\textbf{(a)}$, let us substitute $a=0.10,\ S_{0}=50,000$ into the sales decline function then use this new equation to find $S(1)$ and $S(3)$.

For problem $\textbf{(b)}$, let us substitute $a=0.05,\ S_{0}=80,000$ into the sales decline function then use this new equation to find $S(2)$ and $S(10)$.

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