The fox population in a certain region has a relative growth rate of 5% per year. It is estimated that the population in 2013 was 18,000.
(a) Find a function
n(t) = n0ert
that models the population t years after 2013.
n(t) =
Correct: Your answer is correct.
(b) Use the function from part (a) to estimate the fox population in the year 2021. (Round your answer to the nearest whole number.)
26853
Correct: Your answer is correct.
foxes
(c) After how many years will the fox population reach 27,000? (Round your answer to one decimal place.)
8
Incorrect: Your answer is incorrect.
yr
(d) Sketch a graph of the fox population function for the years 2013–2021.
a 5% growth per year, starting at 18000 means n(t) = 18000 * 1.05^t
Since 1.05 = e^(ln 1.05) that makes
n = 18000 (e^(ln 1.05))^t
n = 18000 e^(ln 1.05 * t)
n = 18000 e^(.0488t)
Now you can answer the other questions.
looks like you missed d)
18000 e^(0.05t) = 27000
e^(.05t) = 1.5
take ln of both sides and use log rules
.05t = ln1.5
t = ln1.5/.05 = appr 8.1 years past 2013
To answer part (c), we need to find the value of t when the population, n(t), reaches 27,000.
Let's plug in the given values:
n(t) = 18,000
n0 = 18,000
r = 5% = 0.05
n(t) = 27,000
We can rearrange the formula n(t) = n0e^(rt) to solve for t in terms of the other variables:
n0e^(rt) = n(t)
e^(rt) = n(t) / n0
rt = ln(n(t) / n0)
t = ln(n(t) / n0) / r
Using the values we have, we get:
t = ln(27,000 / 18,000) / 0.05
t ≈ 8.51
So, after approximately 8.5 years, the fox population will reach 27,000.
To find the function that models the fox population, we can use the formula n(t) = n₀ * e^(r * t), where:
- n(t) represents the population at time t
- n₀ represents the initial population
- r represents the relative growth rate
- t represents the time in years
Given that the relative growth rate is 5%, we can express it as a decimal by dividing it by 100: r = 0.05.
It is also given that the population in 2013 (t = 0) was 18,000, so n₀ = 18,000.
(a) Plugging these values into the formula, the function that models the population t years after 2013 is:
n(t) = 18,000 * e^(0.05 * t)
(b) To estimate the fox population in the year 2021 (t = 2021 - 2013 = 8), we can substitute t = 8 into the function:
n(8) = 18,000 * e^(0.05 * 8) ≈ 26,853 (rounded to the nearest whole number)
(c) To find the number of years it takes for the fox population to reach 27,000, we need to solve the equation:
27,000 = 18,000 * e^(0.05 * t)
To solve for t, we divide both sides by 18,000 and then take the natural logarithm (ln) of both sides:
ln(27,000/18,000) = 0.05 * t
Now we can solve for t by dividing both sides by 0.05:
t ≈ ln(27,000/18,000) / 0.05 ≈ 8.36 (rounded to one decimal place)
Therefore, after approximately 8.4 years, the fox population will reach 27,000.
(d) To sketch a graph of the fox population function for the years 2013-2021, you would create a plot with the years (t) on the x-axis and the fox population (n(t)) on the y-axis. Start with t = 0 (2013) and increment t until t = 8 (2021). Plug each value of t into the function n(t) = 18,000 * e^(0.05 * t) to find the corresponding population value. Plot these points and connect them with a smooth curve.