 # The population (in millions) Ukraine in 2015 is 44.4 and the expected continuous annual rate of change k =−0.006. (a) Find the exponential growth mode facas9 2020-11-26 Answered
The population (in millions) Ukraine in 2015 is 44.4 and the expected continuous annual rate of change k =−0.006.
(a) Find the exponential growth model P = Cekt for the population by letting t = 5 correspond to 2015.
(b) Use the model to predict the population of the country in 2030.
(c) Discuss the relationship between the sign of k and the change in population for the country.
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Given:
The population(in millions) of a country in 2015 is P = 44.4 million,
the expected continuous annual rate of change k = - 0.006 and
t = 5 corresponds to the year 2015.
To find:
(a) The Exponential growth model
(b) To predict the population of the country in 2030 using the given model.
(c) The relationship between the sign of k and the change in population for the country.
(a) To find the Exponential growth model $P=C{e}^{\left(kt\right)}$ for the population by letting t = 5 correspond to 2015. The formula used to find the Exponential growth mode is
$P=C{e}^{\left(kt\right)}$
putting values of p, k & t in the above equation
$44.4=C{e}^{\left(-0.006\left(5\right)\right)}$
$44.4=C{e}^{\left(-0.03\right)}$
$C=44.44{e}^{\left(0.03\right)}$
C=45.752
Therefore,the exponential growth model is
$P={45.752}^{\left(-0.006t\right)}$
(b)Using the above Exponential growth model to predict the population of the country in 2030.
The year 2030 corresponds to t = 20.
Then the population will be,
${P}_{2}030=45.752{e}^{\left(-0.006\left(20\right)\right)}$
$=45.752{e}^{\left(-0.12\right)}$
$=40.578$
(c) The relationship between the sign of k and the change in population for the country:
Here, the sign of k is negative (k < 0). With a positive relationship, these limiting factors increase with the size of the population and limit growth as population size increases. With a negative relationship, population growth is decreases and becomes less limited as it grows.
As we have, k < 0, the population is declining.