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?

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 ??

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!

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.

Thanks for your help! Much appreciated :)!

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

To determine the year when New York will need to change the way they make license plates, we need to calculate the current number of license plates and estimate when that number will exceed the possible combinations using the given population and growth rate.

First, let's calculate the current number of license plates in New York.

We know that each license plate consists of 4 letters followed by 3 digits. Since there are 26 letters in the English alphabet and 10 digits (0-9), we have a total of 26^4 * 10^3 possible combinations.

Therefore, the current number of license plates in New York is: 26^4 * 10^3 = 46,656,000.

Next, let's calculate how many license plates are needed to accommodate the entire population of New York.

The number of license plates needed can be derived from the car-to-resident ratio.
Currently, there are 560 cars for every 1000 residents.

So, the number of license plates needed can be calculated by multiplying the current population by the car-to-resident ratio:

License plates needed = Current population * (Cars per 1000 residents / 1000)
= 12,929,000 * (560 / 1000)
= 7,249,840

Now, let's estimate in which year the number of required license plates will exceed the total combinations possible.

To do this, we need to account for the growth rate of the population. The rate of growth of the population is given as 0.02%.

To estimate the future population in a given year, we need to apply the growth rate formula:
Future population = Current population * (1 + Growth rate)^n

Where "n" is the number of years in the future.

We want to find the year when the required number of license plates exceeds the total combinations, so we set this equation up:

License plates needed = Total combinations possible

7,249,840 = 46,656,000 * (1 + 0.02%)^n

Simplifying the equation, we get:

(1 + 0.02%)^n ≥ 46,656,000 / 7,249,840

Taking the natural log (ln) of both sides, we get:

n * ln(1 + 0.02%) ≥ ln(46,656,000 / 7,249,840)

Finally, solving for "n" gives us the estimated number of years in the future when the required number of license plates will exceed the total combinations possible.

n ≥ ln(46,656,000 / 7,249,840) / ln(1 + 0.02%)

Using a calculator to perform the calculations, we find that n is approximately 7.4 years.

Therefore, New York will need to change the way they make license plates in approximately 7 to 8 years from now.