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

Question

anonymous · Answer

A. f(x)=4(1.15)^x ; growth rate is 5%

B. f(x)=4(1.15)^5x ; growth rate is 115%

C. f(x)=4(2)^x ; growth rate is 200%

D. f(x)=4(2)^x ; growth rate is 5%

zepdrix · Answer

Hey :)

zepdrix · Answer

They want an equation that is `equivalent` to the first one given.
So we'll have to start with that, and manipulate it to get a 5 in the exponent.

zepdrix · Answer

I want to change the 2 in a fancy way, applying some exponent rules,$$\Large\rm \color{orangered}{2}=2^1=2^{(\frac{1}{5}\cdot5)}=\color{orangered}{\left(2^{1/5}\right)^5}$$

zepdrix · Answer

So our equation$$\Large\rm y=4\left[\color{orangered}{2}\right]^x$$can be written as$$\Large\rm y=4\left[\color{orangered}{\left(2^{1/5}\right)^5}\right]^x$$

zepdrix · Answer

Applying exponent rules a little further$$\Large\rm y=4\left(2^{1/5}\right)^{5x}$$gives us the 5 in the exponent like we want.
We need to recalculate the base of this exponential though.

zepdrix · Answer

2^(1/5) is probably 1.15 since that's what is showing up in your answer choices.
Then you need to think about what growth rate 1.15 corresponds to.
5% or 115%?

anonymous · Answer

115?

anonymous · Answer

@zepdrix are you still there?

zepdrix · Answer

115% ?
Yah that sounds right! :)

anonymous · Answer

I already had B selected, I was just double checking. Word problems aren't really my thing.  Thank you! I will fan and give you a medal :)

zepdrix · Answer

yay team \c:/