Question #N251
A certain college had 3,000 students enrolled in 2015. The college predicts that after 2015, the number of students enrolled each year will be 2% less than the number of students enrolled the year before. Which of the following functions models the relationship between the number of students enrolled, \(f(x)\), and the number of years after 2015, x ?
A. \(f(x) = 3,000(0.02)^x\)
B. \(f(x) = 0.98(3,000)^x\)
C. \(f(x) = 3,000(0.98)^x\)
D. \(f(x) = 3,000(1.002)^x\)
Correct Answer is: C
Because the change in the number of students decreases by the same percentage each year, the relationship between the number of students and the number of years can be modeled with a decreasing exponential function in the form \(f(x) = a(1 - r)^x\) where \(f(x)\) is the number of students, a is the number of students in 2015, r is the rate of decrease each year, and x is the number of years since 2015. It's given that 3,000 students were enrolled in 2015 and that the rate of decrease is predicted to be 2%, or 0.02. Substituting these values into the decreasing exponential function yields \(f(x) = 3,000(1 - 0.02)^x\), which is equivalent to \(f(x) = 3,000(0.98)^x\).