Question

From 1953 to 1966 the number of automobiles in the country increased at an annual rate of 6.7 percent. What is the doubling time for growth in automobiles in the US?

Answer 1

Annual increase is 6.7% . If initially N automobiles were there in 1953
After 1 year the no. of automobiles is (106.7/100)*N.
After 2 years i.e, in 1955 it'll be (1.067)*(1.067)*N
Similarly after x years it'll be $$(1.067)^{x} \times N$$
Make it equal to 2N since we need to find the time taken to double.
$$(1.067)^x=2 \implies x=\frac{\ln2}{\ln1.067}=10.7$$
It takes ~11 yrs for the doubling of the automobiles.

Answer 2

Or: \[1.067^{x}=2 => x =\log_{1.067} 2\approx10.7\]

Answer 3

Thank-you to both of you!