4) The function f(x)=2(3)^x represents the growth of a dragonfly population every year in a remote swamp. Rose wants to manipulate the formula to an equivalent form that calculates seven times a year, not just once a year. Which function is correct for Rose’s purpose and what is the new growth rate?

Question

A) f(x)=2(3)^x, growth rate 200%
B) f(x)=2(1.17)^7x, growth rate 17%
C) f(x)=2(3)^x,growth rate 2%
D) f(x)=2(1.17)^x,growth rate 2%

I do not know what to do with this question. I am very confused. I'm thinking it is B or D, but maybe D?
Please explain this to me!! :)

Answer 1

If I understand your question correctly, this is the same as finding the equivalent of an annual rate with new compounding frequencies
the way it stands, f(x) = 2(3)^x
or f(x) = 2( 1 + 2)^x to me represents a growth of 200% compounded annually
so you want
f(x) = 2(1 + r)^7x to be equivalent

2(3)^x = 2(1+r)^7x
3^x = (1+r)^7x
take log of both sides
x log3 = 7x log (1+r)
log 3 = 7log(1+r)
log3 = log( (1+r)^7)
3 = (1+r)^7
take 7th root of both sides
1.16993 = 1+r
r = .16993 or 16.993%

so f(x) = 2(1.16993)^(7x) should do it
which looks like B

Answer 2

I was going to check this:
e.g how much after 3 years
original:
f(3) = 2(3)3 = 54
new one:
f(3) = 2(1.16993)^21 = 53.9992 , not bad
According to B
f(3) = 2(1.17)^21 = 54.067 ,

so it is B