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)

Question

campbell_st · Answer

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

anonymous · Answer

yes

campbell_st · Answer

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...?

campbell_st · Answer

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

anonymous · Answer

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

campbell_st · Answer

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..?

campbell_st · Answer

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

anonymous · Answer

so b=0.0860465116?

anonymous · Answer

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

campbell_st · Answer

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

campbell_st · Answer

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

anonymous · Answer

That is where I get confused....

campbell_st · Answer

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

campbell_st · Answer

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

anonymous · Answer

9.99459=a Is that correct?

campbell_st · Answer

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

anonymous · Answer

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!

campbell_st · Answer

good luck