the fox population in a certain region has a continuous growth rate of 6 percent per year. it is estimated…

the fox population in a certain region has a continuous growth rate of 6 percent per year. it is estimated that the population in the year 2000 was 15000.\n(a) find a function that models the population t years after 2000 (t = 0 for 2000).\nhint: use an exponential function with base e.\nyour answer is p(t) =\n(b) use the function from part (a) to estimate the fox population in the year 2008.\nyour answer is (the answer must be an integer)

the fox population in a certain region has a continuous growth rate of 6 percent per year. it is estimated that the population in the year 2000 was 15000.\n(a) find a function that models the population t years after 2000 (t = 0 for 2000).\nhint: use an exponential function with base e.\nyour answer is p(t) =\n(b) use the function from part (a) to estimate the fox population in the year 2008.\nyour answer is (the answer must be an integer)

Answer

Explanation:

Step1: Determine the general form of the exponential function

The general form of an exponential function for continuous growth is (P(t)=P_0e^{rt}), where (P_0) is the initial population, (r) is the growth rate, and (t) is the time. Given (P_0 = 15000) and (r=0.06), the function is (P(t)=15000e^{0.06t}).

Step2: Calculate the population in 2008

For the year 2008, (t = 2008 - 2000=8). Substitute (t = 8) into the function (P(t)=15000e^{0.06t}). [ \begin{align*} P(8)&=15000e^{0.06\times8}\ &=15000e^{0.48}\ &\approx15000\times1.616178\ &=24242.67 \end{align*} ] Rounding to an integer, we get (24243).

Answer:

(a) (P(t)=15000e^{0.06t}) (b) (24243)