Precalculus question

Question

The fox population in a certain region has an annual growth rate of 4 percent per year. It is estimated that the population in the year 2000 was 29200.

(a) Find a function that models the population  years after 2000 ( for 2000).Your answer is 

(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)__?

Answer 1

If something grows by 4% every year, each year's population is 104% of the year before.

After 1 year, 29200(1.04). After 2 years, 29200(1.04)(1.04), etc.

P(t) = 29200(1.04)^t

The second part is on you.

Go.

Answer 2

Since this is a growth function, begin with P(t) = Ae^kt. The problem stated that when t = 0, P = 2000, so plug this in and you determine that A = 2000 (because anything raised to 0 is one). So P(t) = 2000e^kt. Since the growth rate is 4% per year, in one year the population would have increased by 4% or to 2080. So

2080 = 2000e^k.

Solve for k as follows:

2080/2000 = e^k (by dividing both sides by 2000.

ln(2080/2000) = ln(e^k) = k*ln(e) = k

So your equation becomes P(t) = 2000*e^(ln(2080/2000)t. To determine population at 2008 (8 years from the start plug 8 in for t and you're done!