Question

the fox population in a certain region has an annual growth rate of 7 percent per year. it is estimated that the population in the year 2000 was 15700.
(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 should be an integer)
question help: video

Explanation:

Step1: Identify the growth model

This is an exponential growth problem. The formula for exponential growth is \( P(t) = P_0(1 + r)^t \), where \( P_0 \) is the initial population, \( r \) is the growth rate (in decimal), and \( t \) is time.
Here, \( P_0 = 15700 \), \( r = 0.07 \) (since 7% = 0.07).

Step2: Form the population function

Substitute \( P_0 \) and \( r \) into the formula: \( P(t)=15700(1 + 0.07)^t=15700(1.07)^t \).

Step3: Calculate for 2008

2008 is 8 years after 2000, so \( t = 8 \).
Substitute \( t = 8 \) into \( P(t) \): \( P(8)=15700(1.07)^8 \).
Calculate \( (1.07)^8 \approx 1.718186 \).
Then \( P(8)=15700\times1.718186\approx 15700\times1.718186\approx26975.5202 \).
Round to the nearest integer: \( 26976 \).

Answer:

(a) \( P(t) = 15700(1.07)^t \)
(b) 26976