Question

14. The population of a town is as follows:
Year:
1941 1951 1961 1971 1981 1991
Population in lakhs : 20 24 29 36 46 51
Using interpolation,' estimate the population
increase during the period 1946 to 1976.​

Answer 1

Answer:

Step-by-step explanation:54.777

Answer 2

The estimated increase in population during the period 1946 to 1976 using interpolation is 29 Lakhs.

Step-by-step explanation:

Given:

• The population of a town is given as:
• Year: 1941 1951 1961 1971 1981 1991
• Population in lakhs : 20 24 29 36 46 51

To be found: The population increase during the period 1946 to 1976.​

Formula Used: $y=y_1+(x-x_1)\times \frac{y_2-y_1}{x_2-x_1}$

Solution:

Let $x_1$ be 29 and $x_2$ be 36.

Then $y_1$ will be 1961 and $y_2$ will be 1971.

So, now using the formula of interpolation let us find out the population in 1946,

$y=y_1+(x-x_1)\times \frac{y_2-y_1}{x_2-x_1}\\1946=1961+(x-29)\times \frac{1971-1961}{36-29}\\1946=1961+(x-29)\times \frac{10}{7}\\1946\times7=1961\times7+10x-290\\13622=13727+10x-290\\13622-13727+290=10x\\10x=185\\\therefore x=10.5$

Population in 1946 is 10.5 Lakhs.

Similarly for year 1976,

$y=y_1+(x-x_1)\times \frac{y_2-y_1}{x_2-x_1}\\1976=1961+(x-29)\times \frac{1971-1961}{36-29}\\1976=1961+(x-29)\times \frac{10}{7}\\1976\times7=1961\times7+10x-290\\13832=13727+10x-290\\13832-13727+290=10x\\10x=395\\\therefore x=39.5$

Population in 1976 is 39.5 Lakhs.

Increase in population:

$= 39.5-10.5\\=29$

So, the increase in population during the period 1946 to 1976 is 29 Lakhs.