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
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:
[tex]50000=20000(1.064)^x\\\\2.5=1.064^x\\\\xln(1.064)=ln(2.5)\\\\x=14.7\ years[/tex]
Therefore it would take approximately 14.7 years to reach 50000 members.
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