## HelpMe94 2 years ago What does the Exponential Decay Variables Mean? For Half Lifes Y=Ae^-bX

1. campbell_st

ok so your formula is A is the initial population b is the decay coefficient X is normally the time. $Y = Ae^{-bx}$ then then is the population half of the initial population of A for half life you will have $\frac{Y}{A} = \frac{1}{2}$ so the equation becomes $\frac{1}{2} = e^{-bX}$ you can find - bX by taking the log of both sides of the equation

2. HelpMe94

Another thing is if i know the half life in years = 5715 and the amount after 1000 years is 2 grams then what is the initial amount?

3. HelpMe94

so normal time is the amount of time passed?

4. campbell_st

so then X = 5715 this will help to find the decay constant - b $\frac{1}{2} = e^{-b \times 5715}$ you'll need to solve for b by using logs... the when you have b you will be able to find A given Y = 2 and X = 1000

5. campbell_st

thats correct.

6. HelpMe94

ok so is 1/2 the grams?

7. HelpMe94

also if it is then why is it divided by 1

8. campbell_st

well half life means you end up with 1/2 of what you start with... and because you are given X = 5715 you can calculate b

9. HelpMe94

give me a moment...

10. campbell_st

the calculation is $\ln(\frac{1}{2}) = -b \times 5715$

11. HelpMe94

i see that but what i don't understand is the 1/2

12. campbell_st

half life if you start with 100 gm then 50gm is the half life, 20gm then 10gm is the half life. 3 gm then 1.5 gm is the half life. so its when the initial quantity gets to half its size.

13. campbell_st

so if you really don't need to know the initial quantity if you know how long it takes to get 1/2 life...

14. HelpMe94

oh so is 1/2 just to solve to get b then plug in to the equation for the 1000 years =x to get the answer...

15. campbell_st

thats it... b will be a positive decimal...

16. HelpMe94

ok let me see if this checks out for a second

17. HelpMe94

one thing were did A go?

18. HelpMe94

oh wait i see

19. HelpMe94

y/a is 1/2

20. campbell_st

the ratio of Y/A = 1/2