please help
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) = n0ert
that models the population t years after 2005.
n(t) =
(b) Use the function from part (a) to estimate the fox population in the year 2010. (Round your answer to the nearest whole number.)
foxes
(c) After how many years will the fox population reach 25,000? (Round your answer to one decimal place.)
yr
n(t) = 19000 e^(.07t) , where t is the number of years since 2005
in 2010 , t = 5
n(5) = 19000 e(.07(5)) = 26962
use the method I showed you in your previous post do answer c)
(a) To find a function that models the population, we use the formula for exponential growth:
n(t) = n₀ * e^(rt)
In this case, the relative growth rate is 7% per year, which means r = 0.07. The initial population in 2005 is given as 19,000, so n₀ = 19,000. Thus, the function that models the population is:
n(t) = 19,000 * e^(0.07t)
(b) To estimate the fox population in the year 2010, we substitute t = 2010 - 2005 = 5 into the function:
n(5) ≈ 19,000 * e^(0.07 * 5)
Using a calculator, we can evaluate this expression to get the estimated fox population:
n(5) ≈ 19,000 * e^(0.35)
Round this answer to the nearest whole number to get the estimated fox population in 2010.
(c) To find out how many years it will take for the fox population to reach 25,000, we set the population function equal to 25,000 and solve for t:
19,000 * e^(0.07t) = 25,000
Divide both sides by 19,000:
e^(0.07t) = 25,000 / 19,000
Take the natural logarithm of both sides to isolate the exponent:
ln(e^(0.07t)) = ln(25,000 / 19,000)
The natural logarithm of e to any power is just the power itself:
0.07t = ln(25,000 / 19,000)
Divide both sides by 0.07 to solve for t:
t = ln(25,000 / 19,000) / 0.07
Using a calculator, evaluate this expression to get the approximate number of years it will take for the fox population to reach 25,000. Round your answer to one decimal place.