The current fox population in a protected region is 367 and is decreasing by 8% per year. Find the exponential function f(x) that represents the fox population, where x is the number of years from now. Estimate the fox population of the region 9 years from now.
To find the exponential function, we start with the initial value of the fox population, which is 367. Then, we need to consider that the population is decreasing by 8% per year, so the growth rate is -8%. We use this growth rate to form the base of the exponential function.
The exponential decay function is given by:
f(x) = a(1 - r)^x,
where:
f(x) represents the fox population after x years,
a is the initial value of the population,
r is the growth rate expressed as a decimal,
x is the number of years from now.
In this case, a = 367 and r = -8% = -0.08.
Substituting the values into the exponential decay function:
f(x) = 367(1 - 0.08)^x.
To estimate the fox population 9 years from now, we substitute x = 9 into the exponential function:
f(9) = 367(1 - 0.08)^9.
Calculating this expression yields:
f(9) ≈ 183.99.
Therefore, the estimated fox population in the region 9 years from now is approximately 184.