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Population Size given annual Growth Rate and Time










































Population Growth Function

The population size of a country is expected to grow 4% per annum for the next 10 years.

   • Write an equation which models Population Growth, given

             - initial population

             - growth factor

             - time

Given that the initial population size is now 100,000, what is its population size after 5 years?

Comments for Population Size given annual Growth Rate and Time

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Sep 30, 2012
Population Size
by: Staff


Answer:

Part I

When the rate of change in your data from one period to the next is a constant percentage (%), the function which describes that change is called an exponential function .


      Definitions:


                   t = time (the time could be the number of days, the number of quarters, the number of years, etc.)


                   Growth Factor: (1 + rate of change) is called the “growth factor” if its value is greater than 1.


                   Decay Factor: (1 + rate of change) is called the “decay factor” if its value is less than 1, but greater than 0.


                   (1 + rate of change)t = (1 + rate of change) raised to the “t” power


      An Exponential Function has the form:

                   f(t) = (initial value)*(1 + rate of change)t


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Sep 30, 2012
Population Size
by: Staff


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Part II

      This equation can be applied to your question as follows:

                   P = (initial size of population)*(1 + rate of change) t

                   P = abt

                   P = population of country
                   a = initial size of population
                   b = (1 + rate of change)
                   t = time

                   Note that the growth factor, b = 1 + rate of change
                   (rate of change = growth in 1 time period )

              The final equation can be written as:

                   P = a(1 + rate of change)t

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                   P = calculated size of population = unknown
                   a = initial size of population = 100,000 people
                   rate of change = 4% increase per year (converted to decimal = 0.04)
                   t = time in years = 5 years

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Sep 30, 2012
Population Size
by: Staff


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Part III


              The equation for the growth of the population is:

                   P = a(1 + rate of change)t

                   P = 100000(1 + .04)t

                   P = 100000(1.04)t

              The final equation is: P = 1000000(1.04)t

                   When t = 5 years

                   P = 100000(1.04)5

                   P = 100000(1.2166529024)

                   P = 121665.29024



Final Answer:


                 P = 121,665 people after 5 years (rounded to the whole person)




Thanks for writing.

Staff
www.solving-math-problems.com


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