Question

Estimates of λ or r are often used to determine how rapidly a population is growing (or declining) at various points in time. For a population that is growing exponentially, we can estimate the exponential growth rate (r) using the following equation:

where N(0) is the population size at the beginning of a time period, t is the length of the time period, and N(t) is the population size at the end of the period. In this exercise, we’ll use this technique and the data in the table to examine the growth rate of the world’s human population at different points in time.

Year (c.e.)	Population size	Exponential growth rate (r)
1	170 million	0.00028
400	190 million	?
1400	390 million	?
1800	990 million	?
1925	1.9 billion	?
1965	3.5 billion	?
1980	4.6 billion	?
2000	6.1 billion	?
2016	7.35 billion	(NA)

1 a)Calculate the exponential growth rate (r) for the years shown in the table. For example, from year 1 to year 400, the length of the time period, t, is t= 400 – 1 = 399. The population size at the start of this time period, N(0), equals 170 million, and the population size at the end of this time period, N(t), equals 190 million. If we substitute these values for t, N(0), and N(t) into the equation forrprovided above, we find that

r= [ln(190 million/170 million)]/399 = 0.1112/399 = 0.00028.

b) If the human population continued to grow at the rate you calculated for 2000, how large would the population be in 2050? [Hint: recall from your textbook that in an exponentially growing population, N(t) = N(0) e^rt.]

c) What assumptions did you make in answering Question 2? Based on results for Question 1, is it likely that the human population will reach the size that you calculated for 2050? Why or why not?

Answer 1

ANSWER 1

For Year 400 to the Year 1400,

t= 1400-400=1000

N(0)= 190 million and N(t)= 390 million

Exponential Growth (r)= (390 million/190 million)/1000= 0.00205.

For the Year 1400 to the Year 1800

t= 1800-1400= 400

N(0)= 390 million and N(t)= 990 million

Exponential Growth (r)= (990 million/390 million)/400 = 0.00634

For the Year 1800 to the Year 1925

t = 1925-1800= 125

N(0)= 990 million and N(t)= 1.9 billion

Exponential Growth (r)= (1.9 billion/990 million)/125= 0.0153

For the Year 1925 to the Year 1965

t= 1965-1925= 40

N(0)= 1.9 billion and N(t)=3.5 billion

Exponential Growth (r)= (3.5 billion/1.9 billion)/40= 0.0460

For the Year 1965 to the Year 1980

t= 1980-1965= 15

N(0)= 3.5 billion and N(t)= 4.6 billion

Exponential Growth (r)= (4.6 billion/3.5 billion)/15=0.0876

For the Year 1980 to the Year 2000

t= 2000-1980=20

N(0)=4.6 billion and N(t)= 6.1 billion

Exponential Growth (r)= (6.1 billion/4.6 billion)/20= 0.0663

For the Year 2000 to the Year 2016

t= 2016-2000=16

N(0)= 6.1 billion and N(t)= 7.35 billion

Exponential Growth (r)= (7.35 billion/6.1 billion)/16=0.075

ANSWER 2

N(0)= 6.1 billion

r=0.075

t=2050-2000=50

N(t)= ?

Therefore, according to the formula= N(t)= N(0).ert

so, N(t)= 6.1 billion. e0.075*50

N(t)= 6.1 billion . e3.75

N(t)=6.1 billion . 42.52

N(t)=2.6*1011 (approx. trillion)

ANSWER 3

In question no. 2, all the units were given as calculated in the answer. of the Question No. 1 and only the N(t) i.e., the population size at the end of the time period was to be calculated which was 2050 (as mentioned in the question). So, by using the formula N(t)= N(0).ert calculations were performed.

Yes, it is very likely that the human population will reach the size as calculated for 2050 because there is no planning of the family and it affects the population size. And, if this goes on then it is possible that there no control of the population size.