The fox population in a certain region has a continuous growth rate of 6 percent per year. It is estimated that the population in the year 2000 was 18

shadsiei

shadsiei

Answered question

2021-01-05

The fox population in a certain region has a continuous growth rate of 6 percent per year. It is estimated that the population in the year 2000 was 18900.
(a) Find a function that models the population t years after 2000 (t=0 for 2000). Hint: Use an exponential function with base e.
(b) Use the function from part (a) to estimate the fox population in the year 2008.

Answer & Explanation

cheekabooy

cheekabooy

Skilled2021-01-06Added 83 answers

The fox population in a certain region has a continuous growth rate of 6 percent per year. It is estimated that the population in the year 2000 was 18900.
Calculation:
The general form of an exponential equation is,
P(t)=P0ert
Here, P0=18900
r=6%=6100=0.06
a) Therefore, a function that models the population t years after 2000 is
P(t)=18900e0.06t
b) For 2000, t=0
Therefore for 2008, t=8
Therefore, the estimation of fox population in the year 2008 is
P(8)=18900e0.06(8)
P(8)30544
alenahelenash

alenahelenash

Expert2023-06-14Added 556 answers

(a) We know that the population has a continuous growth rate of 6 percent per year, which can be expressed as a growth factor of 1 + 0.06 = 1.06 per year.
Since we're starting from a population of 18,900 in the year 2000, we have the initial condition P(0) = 18,900.
Using these pieces of information, we can write the exponential function as:
P(t)=P(0)·(1+r)t where r is the growth rate per year.
Substituting the given values, we have:
P(t)=18900·(1+0.06)t
Simplifying further:
P(t)=18900·(1.06)t
Therefore, the function that models the population t years after 2000 is P(t)=18900·(1.06)t.
(b) To estimate the fox population in the year 2008 (8 years after 2000), we can substitute t = 8 into the function we found in part (a):
P(8)=18900·(1.06)8
Calculating the value:
P(8)18900·(1.06)818900·1.59384830,144.43
Therefore, the estimated fox population in the year 2008 is approximately 30,144 (rounded to the nearest whole number).
star233

star233

Skilled2023-06-14Added 403 answers

(a) To model the population growth, we can use the formula for exponential growth:
P(t)=P0×ert
where:
P(t) represents the population at time t,
P0 is the initial population (at t=0),
e is the base of the natural logarithm (approximately 2.71828),
r is the growth rate as a decimal (6 percent = 0.06),
and t is the time in years.
Given that the population in the year 2000 was 18900 (at t=0), we can substitute the values into the equation:
P(t)=18900×e0.06t
(b) To estimate the fox population in the year 2008 (t=8), we can substitute t=8 into the function from part (a):
P(8)=18900×e0.06×8
Now we can evaluate the expression to find the estimated population in the year 2008.
karton

karton

Expert2023-06-14Added 613 answers

Result:
(a)P(t)=18900·e0.06t
(b)30,582
(a) To model the fox population, we can use an exponential function with base e. Let's denote the population at time t years after 2000 as P(t). We know that the continuous growth rate is 6 percent per year, which can be written as a decimal as r=0.06.
Since the base of our exponential function is e, the general form is P(t)=P0·ert, where P0 is the initial population at t=0.
Given that the population in the year 2000 was 18,900, we have P0=18900. Substituting these values into the equation, we get:
P(t)=18900·e0.06t
(b) To estimate the fox population in the year 2008, we need to find the value of P(8) since 2008 is 8 years after 2000. Substituting t=8 into the equation derived in part (a), we have:
P(8)=18900·e0.06·8
To evaluate this expression, we can use the approximate value of e as 2.71828. Calculating this, we find:
P(8)=18900·e0.4818900·1.618742=30582
Therefore, the estimated fox population in the year 2008 is approximately 30,582.

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