# 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

 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 ---------------------------------------------------

 Sep 30, 2012 Population Size by: Staff --------------------------------------------------- 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 ------------------------------------------------------                    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 ---------------------------------------------------

 Sep 30, 2012 Population Size by: Staff --------------------------------------------------- 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