Page 96 - Mathematics Class - XII

PROJECT 2

AIM
Estimation of the population of a particular region/country under the assumptions that there is no migration in or
out of the existing population in a particular year.

PROCEDURE AND OBSERVATION
1. Find the population of a selected region in a particular year.

2. Find the number of births and number of deaths in the existing population in a particular year ‘t’ (say).
Here, P (t) : denotes the whole population of that selected region in a particular year t.

B (t) : denotes the number of births of that selected region in one year between t and t + 1.
D (t) : denotes the number of deaths of that selected region in one year between t and t + 1.

3. Now, establish a relation among them.
P (t + 1) = P (t) + B (t) – D (t) ........ (1)

4. Now, assume

b = Bt () represents the birth rate for the time interval t to t + 1.
Pt ()

d = Dt () represents the death rate for the time interval t to t + 1
Pt ()
Now take the equation (1) again and reform that

P (t + 1) = P (t) + B (t) – D (t)

Bt () Dt
()
P (t + 1) = P (t) 1

Pt () Pt ()
P (t + 1) = P (t) [1 + b – d] ....... (2)

5. Putting t = 0 in equation (2), then we get

P (1) = P (0) [1 + b – d]

Again putting t = 1 in equation (2), then we get
P (2) = P (1) (1 + b – d) or

P (2) = P (0) (1 + b – d) (1 + b – d) = P (0) (1 + b – d) 2

Continuing the above process then we get
P (t) = P (0) (1 + b – d) ....... (3)
t

Here, it is assumed that birth rate and death rate remains the same for consecutive years. P (0) denotes the initial
population. Equation (3) gives the mathematical model for calculation of the population in t year.

94