The population of Mastertown was 23,000 in 2012. Assume that Mastertown's population increases at a rate of 2% per year. Write an equation to model the population of Mastertown (y) based on number of years since 2012 (x).

Answer: The equation to model the population of Mastertownn is y=23,000 (1.02) ^x

The population was 23,000 in 2012→y (0) = 23,000

The population increases at a rate of 2% per year.

If the population of Mastertown is y; and the number of years since 2012 is x, then:

In year 2013, x=2013-2012→x=1 (1 year since 2012), the population would be:

y (1) = y (0) + 2% y (0) →y (1) = 23,000+2% (23,000)

Getting 23,000 common factor:

y (1) = 23,000 (1+2%) →y (1) = 23,000 (1+2/100) →y (1) = 23,000 (1+0.02) →y (1) = 23,000 (1.02)

In year 2014→x=2014-2012→x=2 (2 years since 2012), the population would be:

y (2) = y (1) + 2% y (1) = 23,000 (1.02) + 2%[23,000 (1.02) ]

Getting common factor 23,000 (1.02):

y (2) = 23,000 (1.02) (1+2%) →y (2) = 23,000 (1.02) (1+2/100) →y (2) = 23,000 (1.02) (1+0.02) →

y (2) = 23,000 (1.02) (1.02) →y (2) = 23,000 (1.02) ^2

In year 2015→x=2015-2012→x=3 (3 years since 2012), the population would be:

y (3) = y (2) + 2% y (2) = 23,000 (1.02) ^2+2%[23,000 (1.02) ^2]

Getting common factor 23,000 (1.02) ^2:

y (3) = 23,000 (1.02) ^2 (1+2%) →y (3) = 23,000 (1.02) ^2 (1+2/100) →y (3) = 23,000 (1.02) ^2 (1+0.02) →

y (3) = 23,000 (1.02) ^2 (1.02) →y (3) = 23,000 (1.02) ^3

Then for x years from 2012, the population in Mastertown would be:

y (x) = 23,000 (1.02) ^x

Answer: The equation to model the population of Mastertownn is y=23,000 (1.02) ^x