An initial population of 175 quail increases at an annual rate of 22%. Write an exponential function to model the quail population. What will the approximate population be after 5 years?
a) f(x) = 175(0.22)^x; 473
b) f(x) = (175*0.22)^x; 84,587,005
c) f(x) = 175(22)^x; 901,885, 600
d) f(x) = 175(1.22)^x; 473
The correct answer is d) f(x) = 175(1.22)^x; 473.
To model the quail population, we use the formula for exponential growth:
f(x) = a(1 + r)^x
where a is the initial population, r is the annual growth rate as a decimal, and x is the number of years.
Substituting the given values, we get:
f(x) = 175(1 + 0.22)^x
Simplifying:
f(x) = 175(1.22)^x
To find the approximate population after 5 years, we substitute x = 5:
f(5) = 175(1.22)^5
f(5) ≈ 473
Therefore, the approximate population after 5 years is 473.