The number of stores in a particular chain of coffee bars was 200 in 1990 and began growing exponentially with a growth constant of 0.20 year^-1. In what year would one predict the number of stores to reach 10,000? A) 2000 B) 2005 C) 2010 D) 2015

Question

Quizplus · Accepted Answer

The equation for exponential growth is N(t) = N0 * e^(kt), where N(t) is the number of stores at time t, N0 is the initial number of stores, k is the growth constant, and e is Euler's number.
We are given that N0 = 200 and k = 0.20. We want to find the year, t, at which N(t) = 10,000. So, we can set up the following equation:
10,000 = 200 * e^(0.20t)
Dividing both sides by 200:
50 = e^(0.20t)
Taking the natural logarithm of both sides:
ln(50) = 0.20t
t = ln(50)/0.20 ≈ 11 years
Thus, we predict that the number of stores will reach 10,000 in 1990 + 11 = 2001, which is closest to choice C, 2010.

The Number of Stores in a Particular Chain of Coffee