## anonymous one year ago I am so confused on how to find the exponential model. Please help! Find the exponential growth or decay model y = aebt or y = ae−bt for the population of each country by letting t = 10 correspond to 2010. They give two points and want us to find the exponential model. (2010, 8.6) and (2020, 7.4)

1. campbell_st

so the 2 points are for the one model... (2010, 8.6) and (2020, 7.4) is that correct..?

2. anonymous

yes

3. campbell_st

ok... so there a 2 numbers you need to find, a = the initial population b = the growth constant... so using the 2 points you get 2010 $8.6 = ae^{10b}$ 2020 $7.4 ae^{20b}$ does that make sense...?

4. campbell_st

2020 should read $7.4 = ae^{20b}$

5. anonymous

Yes I understand that. Are you supposed to solve for a next?

6. campbell_st

yes... so looking at the 2020 information you can use index laws for the same base to rewrite it as $7.4 = e^{10b} \times ae^{10b}$ does that make sense..?

7. campbell_st

then making a substitution for the 2010 information $7.4 = 8.6 \times e^{10b}$ now you can solve for b

8. anonymous

so b=0.0860465116?

9. anonymous

sorry, forgot a step. b=-.0150282203. Is that correct?

10. campbell_st

not quite its $\frac{7.4}{8.6} = e^{10b}$ now take the ln of both sides so you can solve for b $\ln(\frac{7.4}{8.6}) = 10b$ now solve... b needs to be a negative... because the population has declined

11. campbell_st

great... so you have a decay model... next you need to substitute the value of b into either equation to find the initial population, a

12. anonymous

That is where I get confused....

13. campbell_st

$8.6 = a \times e^{-0.0150282203 \times 10}$

14. campbell_st

so $\frac{8.6}{e^{-0.0150282203 \times 10}} = a$

15. anonymous

9.99459=a Is that correct?

16. campbell_st

that's what I got... so now you have your model... you can check a and b using the 2nd equation when t = 20 to see if you get 7.4

17. anonymous

Oh my goodness thank you soooo much! For some reason I just could not think to divide.....that last part has had me stumped! Now I can breathe a sigh of relief. You are the best!

18. campbell_st

good luck