The population 01 Egypt in 2002 was 74.224.400 and was 76.141.400 in 2006. Assum

Question

The population 01 Egypt in 2002 was 74.224.400 and was 76.141.400 in 2006. Assume the population of Egypt grew exponentially over tins period. Determine the 4-year growth factor. Determine the 4-year percent change. Determine the 1-year growth factor. Determine the 1-year percent change. Define a function. m, that determines the population of Egypt in terms of the number of years that have elapsed since 2002. t. Assume Egypt's population continued to grow according to this model. Use your graphing calculator to determine how many years it will take for the population of Egypt to double. (Be sure to include at least three decimals in your answer.)

Explanation / Answer

Let the initial population be P0 ant in t years it becomes P(t) exponentially with a growth factor r (yearly), then

P(t) = P0 rt   ... ... ... (1)

In above problem we have,

P0 = 74,224,400 (The starting year 2002)

After t = 4 (i.e. in 2006) the population becomes P(t) | t=4 = 76,141,400

(a) Using these data in equation (1), (Four year growth rate b, t = 1, (one interval of four years) we get

76,141,400 = 74,224,400 b, or,  b = 76,141,400 / 74,224,400

hence, b = 1.0258 (The four year growth factor, taking one interval of 4 years)

(b) The four year percent change = (76,141,400 - 74,224,400) x 100 / 74,224,400

= 2.583 %

(c) One year growth rate r, is achieved using equation (1) with t = 4, we get

76,141,400 = 74,224,400 r4, or,  r4 = 76,141,400 / 74,224,400

hence,    r = 1.0064 (The one year growth factor, taking four intervals of 1 year)

(d) One year percent change = (r -1) x 100 = 0.0064 x 100 = 0.64 %

(e) The function m that determines the population in terms of number of years is P(t) given in equation (1) with yearly growth rate. Therefore,

P0 rt = m (t) = 74,224,400 (1.0064)t   ... .... (2)

(f) Let in t years the population of the Egypt will be doubled, then

m(t) = 2 P0 = P0 (1.0064)t, Therefore, t log (1.0064) = log 2

or, t = log 2 / log (1.0064) =  108.650 years

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