In 1969 the antique automobile club of America had 20,000 members. it grew an average of 6.4% per year. how many years did it take its membership to reach 50,000 members

It would take approximately 14.7 years to reach 50000 members.

An exponential growth is in the form:

y = abˣ;

where y, x are variables, a is the initial value of y and b > 1

Let y represent the number of members after x years.

Since they had initially 20000 members hence a = 20000, also their is a growth of 6.4% per year. hence:

b = 100% + 6.4% = 1.064

Therefore:

y = 20000(1.064)ˣ

to reach 50000 members:

Therefore it would take approximately 14.7 years to reach 50000 members.

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Remember the relation:

Ln(a^x) = x*ln(a)

In 1969, the club had 20,000 members, and it grew an average of 6.4% (or 0.064 in decimal form, this is the one we will use) per year.

This means that in 1970, the club had:

20,000 + (0.064)*20,000 = (1.064)*20,000

In 1971, the club had:

(1.064)*20,000 + 0.064*(1.064)*20,000 = (1.064)^2*20,000 members.

Already you can see a pattern, N years after 1969, the club will have:

M(N) = (1.064)^N*20,000 members

Now we want to find the number N such that:

M(N) = 50,000 = (1.064)^N*20,000

Let's solve this for N.

50,000 = (1.064)^N*20,000

50,000/20,000 =  (1.064)^N

5/2 =  (1.064)^N

Now we apply Ln( ) in both sides

ln(5/2) = ln( (1.064)^N) = N*ln( 1.064)

N = ln(5/2)/ln( 1.064) = 14.8

Then will pass 14.8 years (since 1969) to reach the 50,000 members