Answer :
To determine the number of Starbucks by the year 2015, given that they started with 5 shops in 1982 and grew exponentially at a rate of 21% per year, we can use the formula for exponential growth:
[tex]N = N_0 (1 + r)^t[/tex]
where:
- [tex]N[/tex] is the number of shops at the end of the period.
- [tex]N_0[/tex] is the initial number of shops (5 in 1982).
- [tex]r[/tex] is the growth rate (21% or 0.21).
- [tex]t[/tex] is the number of years from the start year to the year of interest (2015-1982).
Let's calculate step by step:
Initial number of shops:
- [tex]N_0 = 5[/tex]
Growth rate:
- [tex]r = 0.21[/tex] (21% expressed as a decimal)
Number of years ([tex]t[/tex]):
- From 1982 to 2015 is [tex]2015 - 1982 = 33[/tex] years.
Apply the exponential growth formula:
- [tex]N = 5 \times (1 + 0.21)^{33}[/tex]
- [tex]N = 5 \times (1.21)^{33}[/tex]
Calculating [tex](1.21)^{33}[/tex]:
- [tex](1.21)^{33} \approx 194.051[/tex]
Therefore:
- [tex]N = 5 \times 194.051[/tex]
- [tex]N \approx 970.255[/tex]
So, by 2015, there would be approximately 970 Starbucks shops.
