Question

the fox population in a certain region has a continuous growth rate of 5 percent per year. it is estimated that the population in the year 2000 was 28600.

(a) find a function that models the population (t) years after 2000 ((t = 0) for 2000).
your answer is (p(t) =)

(b) use the function from part (a) to estimate the fox population in the year 2008.
your answer is (the answer must be an integer)

Explanation:

Step1: Identify the continuous growth model parameters

$$P_0 = 28600, \quad r = 0.05$$

Step2: Write the continuous growth function

$$P(t) = 28600e^{0.05t}$$

Step3: Determine the value of \(t\) for 2008

$$t = 2008 - 2000 = 8$$

Step4: Calculate the population for \(t = 8\)

$$P(8) = 28600e^{0.05 \times 8} = 28600e^{0.4} \approx 42666.15$$

Step5: Round to the nearest integer

$$P(8) \approx 42666$$

Answer:

(a) \(P(t) = 28600e^{0.05t}\)
(b) \(42666\)