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)

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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)

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

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2020 should read \[7.4 = ae^{20b}\]
Yes I understand that. Are you supposed to solve for a next?
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..?
then making a substitution for the 2010 information \[7.4 = 8.6 \times e^{10b}\] now you can solve for b
so b=0.0860465116?
sorry, forgot a step. b=-.0150282203. Is that correct?
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
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
That is where I get confused....
\[8.6 = a \times e^{-0.0150282203 \times 10}\]
so \[\frac{8.6}{e^{-0.0150282203 \times 10}} = a\]
9.99459=a Is that correct?
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
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!
good luck

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