Question

Given the data shown below, which of the following is the best prediction for
the number of years it will take for the population to reach 200,000?
Year
Population
11,920
2
16,800
3
23,300
4
33,000
5
45,750
6
64,000
O A. 9.41
O B. 12.45
O C. 15.82
O D. 18.14

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