Growth rate question with natural log

Question

In 2010, there were about 246 million vehicles (cars and trucks) and about 308.7 million people in the United States. The number of vehicles grew 15.5 ? % over the previous decade, while the population has been growing at 9.7 ? % per decade. If the growth rates remain constant, when will there be, on average, one vehicle per person?

Answer 1

One vehicle per person means that the number of vehicles and the number of people will be equal.

 

So let's find expressions for the both those numbers and see what happens when we set them equal to each other.

 

Let's let d = the number of decades passed since 2010.

 

And let's work in units of million vehicles and million people to make it easier to write out the equations.

 

By the formula for compound growth, the number of vehicles in millions after d decades will be equal to 246 * (1 + .155)^d

 

And after d decades the number of people in millions will be equal to 308.7 * (1 + .097)^d

 

So set these equal to each other:

246 * (1 + .155)^d = 308.7 * (1 + .097)^d

246 * 1.155^d = 308.7 * 1.097^d

 

Take the natural log of both sides:

ln (246 * 1.155^d) = ln (308.7 * 1.097^d)

ln 246 + d ln 1.155 = ln 308.7 + d ln 1.097

 

Get the d's on one side:

d ln 1.155 - d ln 1.097 = ln 308.7 - ln 246

d (ln 1.155 - ln 1.098) = ln 308.7 - ln 246

 

Combine the ln terms using the rule for logarithm of a quotient:

d ln (1.155 / 1.098) = ln (308.7 / 246)

 

Solve for d:

d = ln (308.7 / 246) / ln (1.155 / 1.098)

 

Do the calculations:

d = 4.49

 

Since d is the number of decades, that means that the number of years is 44.9. 

 

Since the starting year was 2010, the year will be 2054.  The .9 means that it will be 9/10 of the way through that year.