Question

Given the data shown below, which of the following is the best prediction for

Answer 1

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

$y_1 =A _0 e ^{kt _1}\\y_2 = A_0 e^{k t_2}\\(t_1,y_1)=(1,11920)\\(t_2,y_2)=(2,16800)$

Substitute the values

$11920=A _0 e ^{k} ---1\\16800 = A_0 e^{2k} ---2$

Divide 1 and 2

$\frac{11920}{16800}=\frac{e^k}{e^{2k}}\\\frac{11920}{16800}=e^{k-2k}\\ln(\frac{11920}{16800})=-k\\k=-1 ln(\frac{11920}{16800})\\k=0.3432\\A_0=y_1 e^{-k t_1}\\A_0=11920 e^{-0.3432}\\A_0=8457.5238$

The exponential function that passes through the points (1, 11920) and (2, 16800) is$y=8457.5238 e^{0.3432t}$

Now we are supposed to find  the best prediction for  the number of years it will take for the population to reach 200,000

$200000=8457.5238 e^{0.3432t}$

$\frac{200000}{8457.5238}=e^{0.3432t}$

$ln(\frac{200000}{8457.5238})=0.3432t$

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