Answer:
The best prediction for the number of years it will take for the population to reach 200,000 is 9.41
Step-by-step explanation:
Year Population
1 11,920
2 16,800
3 23,300
4 33,000
5 45,750
6 64,000
![y_1 =A _0 e ^{kt _1}\\y_2 = A_0 e^{k t_2}\\(t_1,y_1)=(1,11920)\\(t_2,y_2)=(2,16800)](https://tex.z-dn.net/?f=y_1%20%3DA%20_0%20e%20%5E%7Bkt%20_1%7D%5C%5Cy_2%20%3D%20A_0%20e%5E%7Bk%20t_2%7D%5C%5C%28t_1%2Cy_1%29%3D%281%2C11920%29%5C%5C%28t_2%2Cy_2%29%3D%282%2C16800%29)
Substitute the values
![11920=A _0 e ^{k} ---1\\16800 = A_0 e^{2k} ---2](https://tex.z-dn.net/?f=11920%3DA%20_0%20e%20%5E%7Bk%7D%20%20---1%5C%5C16800%20%3D%20A_0%20e%5E%7B2k%7D%20---2)
Divide 1 and 2
![\frac{11920}{16800}=\frac{e^k}{e^{2k}}\\\frac{11920}{16800}=e^{k-2k}\\ln(\frac{11920}{16800})=-k\\k=-1 ln(\frac{11920}{16800})\\k=0.3432\\A_0=y_1 e^{-k t_1}\\A_0=11920 e^{-0.3432}\\A_0=8457.5238](https://tex.z-dn.net/?f=%5Cfrac%7B11920%7D%7B16800%7D%3D%5Cfrac%7Be%5Ek%7D%7Be%5E%7B2k%7D%7D%5C%5C%5Cfrac%7B11920%7D%7B16800%7D%3De%5E%7Bk-2k%7D%5C%5Cln%28%5Cfrac%7B11920%7D%7B16800%7D%29%3D-k%5C%5Ck%3D-1%20ln%28%5Cfrac%7B11920%7D%7B16800%7D%29%5C%5Ck%3D0.3432%5C%5CA_0%3Dy_1%20e%5E%7B-k%20t_1%7D%5C%5CA_0%3D11920%20e%5E%7B-0.3432%7D%5C%5CA_0%3D8457.5238)
The exponential function that passes through the points (1, 11920) and (2, 16800) is![y=8457.5238 e^{0.3432t}](https://tex.z-dn.net/?f=y%3D8457.5238%20e%5E%7B0.3432t%7D)
Now we are supposed to find the best prediction for the number of years it will take for the population to reach 200,000
![200000=8457.5238 e^{0.3432t}](https://tex.z-dn.net/?f=200000%3D8457.5238%20e%5E%7B0.3432t%7D)
![\frac{200000}{8457.5238}=e^{0.3432t}](https://tex.z-dn.net/?f=%5Cfrac%7B200000%7D%7B8457.5238%7D%3De%5E%7B0.3432t%7D)
![ln(\frac{200000}{8457.5238})=0.3432t](https://tex.z-dn.net/?f=ln%28%5Cfrac%7B200000%7D%7B8457.5238%7D%29%3D0.3432t)
t = 9.41
Hence the best prediction for the number of years it will take for the population to reach 200,000 is 9.41