Question

The population of a region is growing exponentially. There were 30 million people in 1980 (when t=0) and 80 million people in 1990.
(a) Find an exponential model, P(t), for the population (in millions of people) at any time t, in years after 1980. NOTE: Keep at least 4 decimal places in your answer.
(b) What population do you predict for the year 2000? NOTE: Round to one decimal place. Predicted population in the year 2000
(c) What is the doubling time? NOTE: Round to one decimal place. Doubling time =

Answer 1

Answer:

Step-by-step explanation:

Given that

i) population is growing exponentially

ii) P-0 = initial population at 1980 = 30 million

iii) In 1990, i.e. when t =10, P = 80 m

Let us assume

$P(t) = P_0 e^{kt} \\P(t) = 30 e^{kt}$

Since P(10) =80 is given we have

$80 = 30e^{kt}\\e^{k(10)} = 2.667\\k=\frac{ln 2.667}{10} =0.0981$

Hence we get

$P(t) = 30 e^{0.0981t}$

b) For 2000, t = 20

$P(20) = 30  e^{0.0981(20)} \\=213.36$

i.e. 213.4 million

c) Doubling time is whenP = 60

$P(t) = P_0 e^{0.0981t} \\0.0981 t = ln 2\\t =7.065$

In 7.1 years