NY has 560 cars for 1000 residents. The current population of NY is 12,929,000 and rate of growth of the population is 0.02%. NY license plates consists of 4 letters followed by 3 digits. What year will they need to change the way they make license plates?

Answer 1

lots of assumptions here, nevertheless, ...

Assumption #1 : ignore certain restrictions on plate names,
e.g. a lot of 4-letter words would not be allowed.

number of different plates
= 26^4 * 10^3
= 456976000

no. of cars in NY currently = 12929000*(560/1000)
= 7240240

Assumption #2 : the ratio of cars:people remains constant at 560:1000

Assumption #3 : the growth rate is per annum

then 7240240(1.02)^t = 456976000
1.02^t = 63.116
t = log 63.116/log 1.02
= 209.3 years

Assumption #4 : Do you think that in about 200 from now we would still have "cars" with metal plates attached to identify them ??

Answer 2

Hi I got 20, 000 years... The rate is 0.02% which is 0.02/100 ?

then 7240240(1.02)^t = 456976000
1.02^t = 63.116
t = log 63.116/log 1.02
= 209.3 years

How did you get that? Thanks!

Answer 3

you are right, I read the rate incorrectly

so my equation should have been

7240240(1.0002)^t = 456976000
1.0002^t = 63.1161398
log(1.0002^t) = log63.1161398
t(log(1.0002)) = log63.1161398
t = log63.1161398/log1.0002
t = 20727 years

using logs is the only way actually solve the equation.
The other way would be by trial and error, a very time-consuming way.

Answer 4

Thanks for your help! Much appreciated :)!

Answer 5

and in 20K years... Will they still even have cars xD?