I know I can use the population growth formula. here is the question; The fox population in a certain region has a relative growth rate of 7% per year. It is estimated that the population in 2005 was 19,000.
(a) Find a function
n(t) = n0e^rt
that models the population t years after 2005.
n(t) = 1900e^(0.7t)
(b) Use the function from part (a) to estimate the fox population in the year 2010. (Round your answer to the nearest whole number.)
5
foxes
(c) After how many years will the fox population reach 25,000? (Round your answer to one decimal place.)
29 yr
so this is what I got but all 3 of my results were incorrect. can somebody show me step by step? I really am trying to know how to do this example.
Sure, I'll show you step by step how to solve this problem.
(a) To find the function n(t) that models the population, we can use the formula n(t) = n0 * e^(rt), where n0 is the initial population, r is the growth rate as a decimal, and t is the number of years.
Given that the relative growth rate is 7% per year, we can convert it to a decimal by dividing it by 100, so r = 0.07.
The initial population in 2005 is given as 19,000, so n0 = 19,000.
Plugging in these values into the formula, we get: n(t) = 19,000 * e^(0.07t).
(b) To estimate the fox population in the year 2010, we need to substitute t = 2010 - 2005 = 5 (since it is the number of years after 2005) into the function we found in part (a):
n(5) = 19,000 * e^(0.07 * 5).
Evaluating this expression, we get: n(5) = 19,000 * e^(0.35) ≈ 19,000 * 1.421 ≈ 27,016.
So, the estimated fox population in the year 2010 is approximately 27,016.
(c) To find the number of years it takes for the fox population to reach 25,000, we need to solve the equation n(t) = 25,000.
Using the function n(t) = 19,000 * e^(0.07t), we can substitute n(t) = 25,000 into the equation:
25,000 = 19,000 * e^(0.07t).
Dividing both sides by 19,000, we get: e^(0.07t) = 25,000 / 19,000.
Taking the natural logarithm (ln) of both sides to isolate the exponent, we have: ln(e^(0.07t)) = ln(25,000 / 19,000).
Using the property of logarithms, we bring the exponent down: 0.07t * ln(e) = ln(25,000 / 19,000).
Since ln(e) equals 1, we have: 0.07t = ln(25,000 / 19,000).
Dividing both sides by 0.07, we get: t = ln(25,000 / 19,000) / 0.07.
Using a calculator, we find: t ≈ 28.55.
So, it will take approximately 28.55 years for the fox population to reach 25,000.
I hope this explanation helps! Let me know if you have any further questions.
Sure! Let's go through the steps to solve this problem.
(a) First, let's write the population growth formula:
n(t) = n0 * e^(rt)
In this formula, n(t) represents the population at time t, n0 is the initial population, e is Euler's number (approximately 2.71828), r is the relative growth rate (as a decimal), and t is the time in years.
Given that the relative growth rate is 7% per year (or 0.07 as a decimal), and the initial population in 2005 was 19,000, we can substitute these values into the formula:
n(t) = 19,000 * e^(0.07t)
So the correct answer for part (a) should be:
n(t) = 19,000 * e^(0.07t)
(b) To estimate the fox population in the year 2010, we need to find the value of t when t = 2010 - 2005 = 5 years. Substitute this value into the function from part (a):
n(5) = 19,000 * e^(0.07 * 5)
Using a calculator, we can evaluate this expression to get:
n(5) ≈ 19,000 * e^(0.35) ≈ 19,000 * 1.419 ≈ 26,901
Rounding this to the nearest whole number, we get an estimated fox population of 26,901 for the year 2010.
So the correct answer for part (b) is:
Estimated fox population in 2010 ≈ 26,901 foxes
(c) To find out how many years it will take for the fox population to reach 25,000, we need to solve for t in the equation:
25,000 = 19,000 * e^(0.07t)
Divide both sides of the equation by 19,000:
25,000 / 19,000 = e^(0.07t)
Using a natural logarithm (ln) to isolate t, we get:
ln(25,000 / 19,000) = 0.07t
Using a calculator to find the natural logarithm of the fraction, we get:
0.287 = 0.07t
Now, divide both sides of the equation by 0.07:
0.287 / 0.07 ≈ t
Rounding this to one decimal place, we find that it takes approximately 4.1 years for the fox population to reach 25,000.
So the correct answer for part (c) is:
The fox population will reach 25,000 after approximately 4.1 years.
I hope this step-by-step explanation helps you understand how to solve this problem correctly!
Sure! Let's go through the steps to solve this problem.
(a) To find a function that models the population, we use the formula n(t) = n0e^(rt), where:
- n(t) is the population at time t
- n0 is the initial population (in 2005)
- r is the relative growth rate (7% or 0.07 in decimal form)
- t is the time in years
Given n0 = 19,000 and r = 0.07, the function becomes:
n(t) = 19,000e^(0.07t)
(b) To estimate the fox population in the year 2010, we substitute t = 2010 - 2005 = 5 years into the function:
n(5) = 19,000e^(0.07 * 5)
Calculating this, we get approximately 19,000e^(0.35) ≈ 19000 * 1.419 ≈ 26,901
Rounded to the nearest whole number, the estimated fox population in 2010 is 26,901 foxes.
(c) To determine after how many years the fox population reaches 25,000, we need to solve the equation n(t) = 25,000. Using the function from part (a), we can set it up as follows:
25,000 = 19,000e^(0.07t)
To solve for t, we divide both sides of the equation by 19,000:
e^(0.07t) = 25,000/19,000
Then, take the natural logarithm (ln) of both sides to isolate the exponent:
0.07t = ln(25,000/19,000)
Finally, divide both sides by 0.07 to solve for t:
t = ln(25,000/19,000) / 0.07 ≈ 28.9 years (rounded to one decimal place)
So, after approximately 28.9 years, the fox population will reach 25,000.
7% = .07 , not .7 like you had
That was your major mistake.
a) equation:
n(t) = 1900 e^(.07t)
b) if 2005 ---> t = 0
then 2010 ---> t = 5
n(5) = 19000 e^(.07*5) = 26962
I don't understand how you could possible think that 5 foxes would be correct.
there were 19000 foxes and they increased.
Didn't 5 sound rather "unreasonable" to you ?
c) , so we want n(t) = 25000
19000 e^(.07t) = 25000
e^ .07t = 1.315789
take ln of both sides and use log rules
.07t ln e = ln 1.315789
.07t = .274437
t = 3.920.. or appr 3.9 years