If a quantity starts at size P­0 and grows by r% (written as a decimal, r) every time period, then the quantity after n time periods can be determined using either of these relations:

P­n = Po*(1+r)^n

We will correspond n = 0 with the year of 2004, as that is the year for the first piece of data we have. That will make P­0 = 8,569 Starbucks stores. In this problem, we are not given the growth rate, but instead are given that P­2 = 12,440 Starbucks stores.

When n = 2, the explicit equation looks like:

P­2 = P0*(1+r)^2

 We know the value for P­0 = 8,569, so we can put that into the equation:

P­2 = 8569*(1+r)^2

We also know that P­2 = 12,440, so substituting that in, we get

12,440 = 8569*(1+r)^2

8569*(1+r)^2 = 12,440

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click here to see the equation solved for r

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r = 0.205 or 20.5%

Now the equation of the growth is:

P­n = P0*(1+0.205)^n

We need to find n such that Pn = 2P0 that is:

2P0 = P0*(1+0.205)^n

P0*(1+0.205)^n = 2P0 divide both sides by P0

1.205^n = 2

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click here to see the equation solved for n

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n = 3.72 years

The doubling time is 3.72 years.