Answer: One Vehicle per Person in US in 1989

[Jacobpm64]

In 1980, there were about 170 million vehicles (cars and trucks) and about 227 million people in the United States. The number of vehicles has been growing at 4% a year, while the population has been growing at 1% a year. When was there, on average, one vehicle per person?

hmm, I think since we're looking for there to be one vehicle per person, then the number of people and the number of vehicles need to be equal. I'll set the two equations equal to each other.

P (in millions)
P = 170(1.04)^t <----- for vehicles
P = 227(1.01)^t <----- for people

Setting them equal to each other because we're looking for one vehicle per person...
170(1.04)^t = 227(1.01)^t
ln[170(1.04)^t] = ln[227(1.01)^t]
ln170 + ln1.04^t = ln227 + ln1.01^t
ln170 + tln1.04 = ln227 + tln1.01
tln1.04 - tln1.01 = ln227 - ln170
t(ln1.04 - ln1.01) = ln227 - ln170
t = (ln227 - ln170) / (ln1.04 - ln1.01)
t = 9.87864 years

So,
1980 + 9.87864 = 1989.88

Is this correct?

 

[Astronuc]

Certainly looks correct. One could put the answer back into the equations to see if the curves defined by equations intersect, the P's equal at the same time.