Question

A national census is mandated by the Constitution of Pakistan to be held every ten years. After the independence of Pakistan in 1947, the first census took place in 1951. Figure 1 reflects the increase in population of Pakistan during last four national census. The increase between 1972 and 2017 could be approximated by a linear function. If in 1972 the population was 65.309 million and as per the census data of 2017 the population was 207.774 million;

a) Determine the estimation function P = f (t), where P equals the population of Pakistan and t equals time measured in years since 1972.
b) Interpret the slope and P intercept
c) According to this function, what will be the estimated population of Pakistan in 2027 census?

Answer 1

a)
`P(t) = 3166t + 65309`

b) The slope of 3166 means that each year, the population increases, on average, by 3166 million people.

The P-intercept of 65309 means that the population of Pakistan in 1972 was of 65309 million people.

c) The estimated population of Pakistan in 2027 will be of 239 439 million people.

Explanation:

Linear function:

In this question, the linear function for the Population in Pakistan in t years after 1972 is given by the following equation:

`P(t) = mt + P(0)`

In which P(t) is given in millions of people.

In which m is the slope, that is, the average number by which the population changes each year, and P(0) is the population in 1972.

a) Determine the estimation function P = f (t), where P equals the population of Pakistan and t equals time measured in years since 1972.

In 1972, the population was of 65.309 million. This means that
`P(0) = 65.309`.

So

`P(t) = mt + 65.309`

In 2017, the population was 207.774 million.

2017 - 1972 = 45

This means that
`P(45) = 207.774`. So

`207774 = 45m + 65309`

`45m = 207774 - 65309`

`m = ((207774 - 65309))/(45)`

`m = 3166`

So

`P(t) = 3166t + 65309`

b) Interpret the slope and P intercept

Following the explanation before item a:

The slope of 3166 means that each year, the population increases, on average, by 3166 million people.

The P-intercept of 65309 means that the population of Pakistan in 1972 was of 65309 million people.

c) According to this function, what will be the estimated population of Pakistan in 2027 census?

2027 - 1972 = 55

So this is

`P(55) = 3166*55 + 65309 = 239 439`

The estimated population of Pakistan in 2027 will be of 239 439 million people.