Question

homework assignment 6.7 exponential and logarithmic mode
due friday by 11:59pm points 10 submitting an external tool
homework assignment 6.7 exponential and logarithmic models
score: 8.17/10 answered: 9/10
question 10
the fox population in a certain region has a continuous growth rate of 7 percent per year. it is estimated that the population in the year 2000 was 9200.
(a) find a function that models the population t years after 2000 (t = 0 for 2000).
hint: use an exponential function with base e.
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: Recall the continuous - growth formula

The formula 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 in years.
Given \(P_0 = 9200\) (population in 2000) and \(r=0.07\) (7% growth rate, so \(r = 7\div100=0.07\)).
So the function \(P(t)=9200e^{0.07t}\).

Step2: Calculate \(t\) for the year 2008

For the year 2008, \(t = 2008 - 2000=8\).
Substitute \(t = 8\) into the function \(P(t)=9200e^{0.07t}\).
We get \(P(8)=9200e^{0.07\times8}=9200e^{0.56}\).
Using a calculator, \(e^{0.56}\approx1.75067\).
Then \(P(8)=9200\times1.75067 = 16106.164\approx16106\) (rounded to the nearest integer).

Answer:

(a) \(P(t)=9200e^{0.07t}\)
(b) \(16106\)