a Do some research and find a city that has experienced popu

a) Do some research and find a city that has experienced population growth. Determine its population on January 1st of a certain year. Write an exponential function to represent the city\'s population, y, based on the number of years that pass, x, after a period of exponential growth. Describe the variables and numbers that you used in your equation.

b) Find another city whose population starts larger than the city in part (a), but that during this same time experienced population decline. Deternine its population for January 1st of the same year you picked for part (a). Write an exponential function to represent the city\'s population, y, based on the number of years that pass, x, after a period of population decline. Describe the variables and numbers that you used in your equation.

c) Explain the similaritis and differences between your equations in (a) and (b).

d) During what year will the population of city (a) first exceed that of city (b)? Show all of your work and expalin your steps.

e) During what year will the population of city (a) be at least twice the size of the population of city (b)? Show all your work and explain your steps.

Solution

a) Lets asssume population of city A is 25 million in the year 2000 on 1st Jan.Let the growth factor r = 10%

After x years , the population can be given by the equation: y = A(1+r)^x

At x =0 ( i.e. on 1st Jan 2000) ; y = 25 million

y = 25(1+0.1)^x -----> y = 25(1.1)^x

Lets find population after 5 years i.e. on 1st Jan 2005: x =5

y = 25(1.1)^5 = 40.26 million population

b) Lets take another city B withpopulation 40 million on 1sy Jan 2000.The population of the city decreases by the

decay factor r= 20%.After x years, the population of the city can be gievn by the exponential equation:

y = A(1-r)^x = 40(1-0.2)^x = 40(0.8)^x

y = 40(0.8)^x

After x= 5yrs i.e.on 1st Jan2005 , population of city cane given as :

y = 40(0.8)^5 = 13.10 million

c) y = 25(1.1)^x ;   y = 40(0.8)^x

Both the equations are exponential equations where intial quatity increases/ decreases by a factor.D

However the difference lies in the fact that one is a growth equation , other one is a decay equation

d) During what year will the population of city (a) first exceed that of city (b)?

25(1.1)^x = 40(0.8)^x

(1.1/0.8)^x = 40/25

1.375^x =1.6

xln(1.375) = ln(1.6) ----> x = 1.475

In the second year i.e. 2002 population of ciyA exceeds the population of cityB