31. The number of Starbucks stores worldwide has increased sub- n- stantially si

Question

31. The number of Starbucks stores worldwide has increased sub- n- stantially since 1991 (see Table 42). iS rt all Table 42 Numbers of Starbucks Stores Worldwide Number of he Year Stores 1991 1993 1995 1997 1999 2001 2003 116 272 676 1412 2135 4709 7225 Source: Starbucks Corporation Let n f(t) be the number of Starbucks stores worldwide at t years since 1990. a. Find an exponential equation of f b. What is the percentage rate of growth of Starbucks stores? c. Find f(4). What does it mean in this situation? d. Use f to estimate the number of stores in 2008. At th end of 2008, there were 15,256 stores (after 500 store

Explanation / Answer

31.a. Let the exponential equation of f be f(t) = a bt, where a, b are arbitrary real numbers. In 1991, when t = 0, we have 116 = a so that f(t) = 116bt. Further, in 2003, when t= 12, we have 7225 = 116*b12 or, b12 = 7225/116 so that b =(7225/116)1/12 = 1.41101508. Thus, f(t) = 116(1.41101508)t.

b. In 12 years, the no. of Starbucks stores has grown from 116 to 7225 so that the percentage rate of growth in the no. of Starbucks stores from 1991 to 2003 is [(7225-116)/116]*100 =6128.45 %.

c. f(4) =116(1.41101508)4 = 116*3.963935877= 460 ( on rounding off to the nearest whole number). It means that the no. of Starbucks stores in 1995, as per the exponential model, is460.

d. In 2008, t = 17 so that, the no. of Starbucks stores in 2008, as per the exponential model, is116(1.41101508)17 = 116*348.3678245 = 40411.

Note:

The differences mean that the data does not exactly depict an exponential growth.

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