Question

The population P of Texas (in thousands) from 1991 through 2000 can be modeled by: P=16,968e^0.019t

where t=1 represents the year 1991. According to this model, when will the population reach 22 million?

Answer 1

P is the population in thousands, so how do we represent 22 million?

Answer 2

22,000?

Answer 3

For example, if the population was 1000, then P = 1

if the population was 2000, then P = 2

etc etc

Answer 4

good, multiply by 1000 to jump from thousands to millions

Answer 5

so P=22,000?

Answer 6

correct

Answer 7

that means you plug in P = 22000 and solve for t

Answer 8

P=16,968e^(0.019t)

22000=16,968e^(0.019t)

what's the next step?

Answer 9

divide by 22000?

Answer 10

that'll move over the 22000 to the right side

Answer 11

we want to move over things that aren't t to the left side to isolate t

Answer 12

I don't know how to do that haha

Answer 13

16,968e^(0.019t) means 16,968 times e^(0.019t)

Answer 14

so we first move over the 16,968 by undoing the multiplication

Answer 15

so you would divide by 16, 968 then?

Answer 16

exactly

Answer 17

giving you?

Answer 18

1.296

Answer 19

I'm getting 1.2965582272513, so that's roughly the same

Answer 20

so that means e^(0.019t) = 1.2965582272513

what's next?

Answer 21

Would you then figure out what e^0.019 is?

Answer 22

what undoes exponentiation?

Answer 23

take the log?

Answer 24

correct, specifically the natural log (since that's base e)

Answer 25

applying the natural log to both sides (ln...lowercase LN) gives

e^(0.019t) = 1.2965582272513

ln(e^(0.019t)) = ln(1.2965582272513)

0.019t*ln(e) = ln(1.2965582272513)

0.019t*1 = ln(1.2965582272513)

0.019t = ln(1.2965582272513)

hopefully you see which log rules I'm using above. If not, then let me know

Answer 26

If you don't have any questions on that chunk of work, what's next?

Answer 27

so then you would divide by 1.29etc to both sides?

Answer 28

no

Answer 29

you need to isolate t

Answer 30

0.019t is the same as 0.019 times t

Answer 31

you move that 0.019 over to isolate t by undoing whatever is being done to 0.019 and t

Answer 32

sorry my internet connection is really bad right now. You would divide by .019 to isolate t by itself.

Answer 33

correct

Answer 34

divide both sides by 0.019

Answer 35

0.019t = ln(1.2965582272513)

t = ln(1.2965582272513)/0.019

t = ???

Answer 36

13.669117

Answer 37

I'm getting the same

Answer 38

so 13.7 years approximately. add that to 1991 and you'd get 2004.7?

Answer 39

round that to the nearest whole year to get t = 14

so the population will happen in the middle of year 13, but it's definitely over 22 million on year 14

Answer 40

I'd say 1991 + 14 = 2005

Answer 41

So between 2004 and 2005

Answer 42

yeah exactly

Answer 43

thanks!

Answer 44

if they want a whole year, then go for 2005 since 2004 will fall short of 22 million

Answer 45

yw

Answer 46

oh wait

Answer 47

t = 1 represents 1991

Answer 48

so you add the result to 1990 not 1991

Answer 49

1990 + 13 = 2003
1990 + 14 = 2004

so it's in between 2003 and 2004 (go with 2004 if they want a whole numbered year)