Question

If t is in years since 1990, one model for the population of the world, P, in billions, is P=40/1+11e^-0.08t

a) What does this model predict for the maximum sustainable population of the world? Enter the exact answer.

This model predicts that t is very large, the population is ________ billion.

b) According to this model, when will the earth's population reach 10 billion? 39.9 billion?

Round your answer to the nearest year.
The population of the world should be 10 billion in ______.
The population of the world should be 39.9 billion in ______.

Answer 1

not sure what max sustainable means for a) but i know b)

Answer 2

first, substitute 10 for P. your goal is to isolate t

Answer 3

to do this.....
P=40/1+11e^-0.08t

10=40/1+11e^-0.08t (now take denominator to other side by multiplying it)

10(1+11e^-0.08t)=40 (now divide 40 by 10 to simplify)

1+11e^-0.08t=4 (now subtract 1 from 4 to simplify more)

11e^-0.08t=3 (now divide by 11 to simplify)

e^-0.08t=3/11 (now that e is alone, multiply both sides of equation by ln)

ln(e^-0.08t)=ln(3/11) (ln and e cancel neach other out and the exponent on e comes down)

-0.08t=ln(3/11) (use your calculator to solve for ln (3/11) and divide it by -0.08 to find t)

-0.08t=-1.29928
t=16.24 years (round to 16 years)

repeat same process for 32 billion but instead of P=10, do P=32

Answer 4

make sense?

Answer 5

Alright, for a), I figured it out.
It's just 40.

For b) I put 16 in and found P=39.9 to be 76 years, but it's telling me both of them are wrong.

Answer 6

no for b, sub 10 in for p and t should equal 16

Answer 7

because its asking for 10 billion

Answer 8

Nevermind, I figured that first part. It's years after 1990, so +16 is 2006.

Hm... but I guess I did something wrong for the second part.

Answer 9

second part?

Answer 10

39.9 = 40 / 1+11 e^0.08t
1596 = 1+11 e^0.08t
1597 = 11 e^0.08t
ln(145.18) = (e ^0.08t)ln
4.98 = 0.08t
t = 62.22 = 62 years

What did I do wrong?

Answer 11

at the beginning, take the denominator to the other side then divide 40 by 39.9

Answer 12

39.9 = 40 / 1+11 e^0.08t
40 (1+11e^0.08t) = 39.9
1+11e^0.08t = 0.9975
11e^0.08t = -0.0025
ln(e^0.08t) = (-0.0025/11)ln
0.08t = -8.39
t = 104.875 = 105

Okay, got it!
Thanks a lot!

Answer 13

no problem