Question

Annual sales for a fast food restaurant are $650,000 and are Inareasing at a rate of 4% per year.Use an exponential functon to find the annual sales after 7 years.

Answer 1

We can write a function S(t), where S are the annual sales and t is the number of years from now.

Then, P(0) = 650000, as this is the starting point.

If the sales increase at 4% per year, then P(1) can be calculated as:

$\begin{gathered} S(1)=S(0)+\frac{4}{100}\cdot S(0) \\ S(1)=1\cdot S(0)+0.04\cdot S(0) \\ S(1)=(1+0.04)\cdot S(0) \\ S(1)=1.04\cdot S(0) \\ S(1)=1.04\cdot650000 \end{gathered}$

We then can find the sales for the second year and generalize the function:

$\begin{gathered} S(2)=1.04\cdot S(1)=1.04\cdot(1.04\cdot S0))=1.04^2\cdot S(0)=1.04^2\cdot650000 \\ S(t)=1.04^t\cdot650000 \end{gathered}$

Now, we can find the value of the sales after 7 years, that corresponds to the the value of S(7):

$S(7)=1.04^7\cdot650000\approx1.32\cdot650000=855355$

Answer: the annual sales after 7 years are expected to be $855,355