The fox population in a certain region has a continuous growth rate of 9 percent per year. It is estimated that the population in the year 2000 was 23400.
(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)
(a) To find a function that models the population t years after 2000, we can use the formula for exponential growth:
P(t) = P(0) * (1 + r)^t
Where P(t) is the population at time t, P(0) is the initial population, r is the growth rate as a decimal, and t is the number of years after the initial time.
In this case, the initial population in the year 2000 was 23,400, and the growth rate is 9% per year, or 0.09 as a decimal. So, the function that models the population t years after 2000 is:
P(t) = 23,400 * (1 + 0.09)^t
Therefore, the answer is P(t) = 23,400 * 1.09^t.
(b) To estimate the fox population in the year 2008, we can substitute t = 8 into the function we derived in part (a):
P(8) = 23,400 * 1.09^8
P(8) ≈ 23,400 * 1.999343
P(8) ≈ 46,812.6
Rounding to the nearest integer, we estimate the fox population in the year 2008 to be 46,813.
To find a function that models the population, we can use the formula for exponential growth:
P(t) = P0 * (1 + r)^t
Where:
P(t) represents the population at time t,
P0 is the initial population (in the year 2000),
r is the growth rate as a decimal (9% = 0.09),
t is the number of years after 2000.
(a) So, in this case, P0 = 23400 and r = 0.09. Therefore, the function that models the population t years after 2000 is:
P(t) = 23400 * (1 + 0.09)^t
Simplifying this equation, we get:
P(t) = 23400 * (1.09)^t
This is the function that models the population.
(b) To estimate the fox population in the year 2008 (8 years after 2000), we substitute t = 8 into the function:
P(8) = 23400 * (1.09)^8
Calculating this expression, we find:
P(8) ≈ 23400 * (1.09)^8 ≈ 48314
Therefore, the estimated fox population in the year 2008 is approximately 48,314 individuals.
P(t) = 23400 * 1.09^t
Now crank it out