Question

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

Answer 1

Answer:

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