if city A's population is 5000 with a yearly growth rate of 4% and city B has a population of 8000 with a yearly growth rate of 2% when will city A surpass city B?
are the answers:
City A13329.182
City B 13124.849
25 years
???
+23252 City A: population = 5000(1.04)n
City B: population = 8000(1.02)n
These will be equal when: 5000(1.04)n = 8000(1.02)n
Find the log of both sides: log( 5000(1.04)n ) = log( 8000(1.02)n )
Rewriting: log( 5000 ) + log( 1.04n ) = log( 8000 ) + log( 1.02n )
Rewriting: log( 5000 ) + n · log( 1.04 ) = log( 8000 ) + n · log( 1.02 )
Rearranging: n · log( 1.04 ) - n · log( 1.02 ) = log( 8000 ) - log( 5000 )
Factoring: n ( log( 1.04 ) - log( 1.02 ) ) = log( 8000 ) - log( 5000 )
Dividing: n = [ log( 8000 ) - log( 5000 ) ] / [ log( 1.04 ) - log( 1.02 ) ]
Using a calculator, n = 25.
Also: 5000(1.04)25 = 13329
8000(1.02)25 = 13125
