Answer :
To solve the problem of determining how many Starbucks shops existed in 2015, given that there were 5 shops in 1982 with an annual exponential growth rate of 21%, we can use the formula for exponential growth:
[tex]N = N_0 \times (1 + r)^t[/tex]
Where:
- [tex]N[/tex] is the final amount (number of shops in 2015).
- [tex]N_0[/tex] is the initial amount (number of shops in 1982), which is 5.
- [tex]r[/tex] is the growth rate (expressed as a decimal), which is 0.21 for 21%.
- [tex]t[/tex] is the time period in years between 1982 and 2015.
First, calculate the time period [tex]t[/tex]:
[tex]t = 2015 - 1982 = 33[/tex]
Now apply these values to the formula:
[tex]N = 5 \times (1 + 0.21)^{33}[/tex]
[tex]N = 5 \times (1.21)^{33}[/tex]
Calculating [tex](1.21)^{33}[/tex], you get approximately 343.5.
Therefore:
[tex]N \approx 5 \times 343.5[/tex]
[tex]N \approx 1717.5[/tex]
Thus, there were approximately 1,718 Starbucks shops in 2015, considering natural rounding.
