The fox population in a certain region has an annual growth rate of 6 percent per year. It is estimated that the population in the year 2000 was 13600.

Question

(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

As you can see, it says that the population is growing 6% per year. We are always getting the 6% always from the previous year, not from the original year and that make this exponential growth. Therefore our function is exponential.

This is how it goes:

for t =0, (which is the year 2000)

P(0) = 13600

for t=1

P(1) = 13600 + 6%(13600)

if we say r = rate then

P(1) = 13600 + r (13600) = 13600 (1 + r)

for t=2

P(2) =(previous year pop.) + r(previous year pop.)

P(2) = [13600(1 +r)] + r[(13600(1 + r)]. Factor out 13600(1 + r):

P(2) = 13600(1+r)(1+r) =13600(1+r)².

for t=3, you'll see that:

P(3) = 13600(1+r)³

Therefore our function should be:

(a) P(t) = 13600(1 + r)^t, where r=0.06

or P(t)=13600 (1.06)^t

2008 is 8 years from year 2000, Therefore here, t=8:

(b) P(8) = 13600 (1.06)⁸ = 21676

*Take note that we have to round off to the nearest whole number for the population because we can't have a fraction of a fox.

Answer 2

This is an exponential growth problem so use the formula p(t) = A(r)^t where A is initial population, r is growth rate which in this problem is ( 1.00 +.06 =1.06) and t is years which gives us

P(t) = 13600 (1.06)^8= 21,676