Helped with a growth rate problem?

Question

The population of a city triples every 4 years. It starts with a population of 100. What is the anual growth rate of this city?

Heres what i said:

A=100(3)^(n/4)
What I did to solve for the growth rate was look at 3^(n/4) as the 100 is irrelevant. I then made n=1
However i do not think that this is the right way to approch this. Can anyone point me in right way?
Dont need to solve... just maybe tell me where you would start

ignoramus · Accepted Answer

You are correct.

If the population after 4 years is 3 times the original population, then the yearly increase is simply 3^¼ = 1.3161.

Check it out : call the original population 100 for convenience, then

Year 1 : 100 (1.3161) = 131.6

Year 2 : 131.6 (1.316) = 173.2

Year 3 : 173.2 (1.3161) = 228.0

Year 4 : 228.0 (1.3161) = 300

[ It is simply a geometric sequence : a, ar, ar², ar³ . . . . etc

Since you know that the fifth term, ar^4 = 3a,

then r^4 = 3, so r = 3^¼ ]

gile · Answer

A = 100*3^(n/4)
Anual growth rate = dA/dn

ln(A) = ln(100) + (n/4)ln(3)
(1/A)(dA/dn) = (1/4) ln(3)
dA/dn = (A/4) ln(3)
dA/dn = 25 ln(3)*3^(n/4)

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Helped with a growth rate problem?

2 Answers