Question

Use Lagrange’s formula and estimate from the following data the number of workers getting income not exceeding Rs. 26 per month.

Income not exceeding (₹)	15	25	30	35
No. of workers	36	40	45	48

Answer 1

Here the intervals are unequal.

By Lagrange’s In-terpolation formula we have,

x₀ = 15

x₁ = 25

x₂ = 30

x₃ = 35

y₀ = 36

y₁ = 40

y₂ = 45

y₃ = 48 and x = 26.

y = `"f"(x) = ((x_ - x_1)(x - x_2)(x - x_3))/((x_0 - x_1)(x_0 - x_2)(x_0 - x_3)) xx y_0 + ((x - x_0)(x - x_2)(x - x_3))/((x_1 - x_0)(x_1 - x_2)(x_1 - x_3)) xx y_1 + ((x - x_0)(x - x_1)(x - x_3))/((x_2 -x_0)(x_2 - x_1)(x_2 - x_3)) xx y_2 + ((x - x_0)(x - x_1)(x - _2))/((x_3 - x_0)(x_3 - x_1)(x_3 - x_2)) xx y_3`

y = `((26 - 25)(26 - 30)(26 - 35))/((15 - 25)(15 - 30)(15 - 35)) xx 36 + ((26 - 15)(26 - 30)(26 - 35))/((25 - 15)(25 - 30)(25 - 35)) xx 40 + ((26 - 15)(26 - 25)(26 - 35))/((20 - 15)(30 - 25)(30 - 35)) xx 45 + ((26 - 15)(26 - 25)(26 - 30))/((35 - 15)(35 - 25)(35 - 30)) xx 48`

= `((1) xx (-4) xx (-9))/((-10) xx (-15) xx (-20)) xx 36 + (11 xx (-4) xx (-9))/(10 xx (-5) xx (-10)) xx 40 + (11 xx 1 xx (-9))/(10 xx (-5) xx (-10)) xx 45 + (11 xx 1 xx (-4))/(20 xx 10 xx 5) xx 48`

= `(36 xx 36)/(-3000) + (11 xx 36 xx 40)/500 + (-99 xx 45)/(-375) + (-44 xx 48)/1000`

= `1296/(-3000) + 15840/500 + 4455/375 - 2112/1000`

= `- 0.432 + 31.68 + 11.88 - 2.112`

= 43.56 – 2.544

= 41.016

∴ Required No.of workers = 42 Persons (approximately)