Question

The function f(x) = 20(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?

Answer 1

So I know that for a year her growth rate is the f(x)=20(2)^x

Answer 2

@mathmale

Answer 3

By f(x) = 20(2)x, do you mean\[f(x) = 20(2)^x?\]

Answer 4

Yes sir!

Answer 5

I'm uncertain what you mean by your phrase "calculates five times per year." What does that mean to you? Calculate ... what?

Answer 6

I am assuming that the function given is representing the formual for the growth rate of once a year..so she wants one that calculates five years?

Answer 7

Let's compromise and express this as it is normally expressed: "rate of annual growth."

Answer 8

Okay! Would you like me to write out the answer choices just so you can catch a glimpse of what exactly their needs are?

Answer 9

Yes, but first let's discuss this problem a bit further. If, instead of typing\[f(x) = 20(2)^x\] we were to type \[f(x) = 20(2)^{5x}\]

Answer 10

... how do you think the behavior of the original function would be modified?

Answer 11

f(x) = 4(1.15)x; growth rate is 5%

f(x) = 4(1.15)5x; growth rate is 115%

f(x) = 4(2)x; growth rate is 200%

f(x) = 4(2)x, growth rate is 5%

Here are the choices, the numbers/letters after the parenthesis are ^. I believe the behvaior of the function would be modified by increasing the outcome?

Answer 12

The outcomes would be dramatically larger were we to replace that \[2^x\] by \[2^{5x}\]

Answer 13

If you think about what we've learned so far, could you now make an educated guess regarding which of the four possible answer choices is the correct one?

Answer 14

Note that the problem you've typed in begins with a '20,' whereas all of the answer choices begin with a '4'.

Answer 15

I sort of believe it would be b. Just because we added the 5x.

Answer 16

That's a really high growth rate though

Answer 17

Actually, you/we inserted the multiplier '5'...we did not 'add' the '5x'.
In this case I'm unfortunately not clear on how we progressed from 20(2)^x to 4(x)^x. We have to account for that in some way.

Answer 18

I'm really not sure either. I'm confused as to how one can just manipulate a formula

Answer 19

As much as I'd like to help further, I don't yet see what was done with that '20'.

I do see that if we start with f(x) = 20(2)^x, and insert the multiplier 5, we get
\[f(x)=20(32)^x\]

Answer 20

Manipulating a formula is perfectly legal and appropriate. What's not clear to me, and apparently not clear to you, is the significance of inserting that multiplier, '5'.

Answer 21

Is it quite possible that the 20 is representing the amount of butterflies already? And I know that it's legal and appropritate, I'm just confused extrememly on how they're manipulating this problem is what I meant.

Answer 22

Yes, it's true that the '20' in the original problem statement represents the original number of butterflies. Note that if x=0, 2^x = 2^0 =1 =f(0)=initial value of f(x).
Were we to insert that factor of 5, the resulting formula would look like \[20(2)^{5x}=20(32)^x\] which would grow a LOT faster.

Answer 23

a lot faster than the original formula would, I mean.

Answer 24

Right, so now I see how you get the 20(32)^x.

Answer 25

Unfortunately, I share your uncertainty in regard to how to manipulate this last expression so that it resembles one of the four possible answers.

Answer 26

All I can suggest, at this point, is that you double check to ensure you've copied the problem statement correctly.

Answer 27

In your shoes, I'd move on to a new problem and return to this one only if you can and if you have time. Choose whichever answer you think is best, based upon our discussions.

I see you have indeed copied the problem down correctly.

Wish I could help you take this problem all the way to a satisfactory, definite conclusion

Answer 28

This is sadly the only question I have left to answer.

Thank you for attempting to help, i highly appreciate it.

Do you want me to tag some others to see if they can help us out this slump?

Answer 29

Yes, I think you should. I look forward to working with you again in the near future (if not tonight).

Answer 30

@SolomonZelman Do you have any idea how to do this problem?

And most likely next school year! This is my final assignment needed for this years math course. :)