Question

Using interpolation estimate the output of a factory in 1986 from the following data.

Year	1974	1978	1982	1990
Output in 1000 tones	25	60	80	170

Answer 1

Here the intervals are unequal.

By Lagrange’s in-terpolation formula we have,

x₀ = 1974

x₁ = 1978

x₂ = 1982

x₃ = 1990

y₀ = 25

y₁ = 60

y₂ = 80

y₃ = 170, and x = 1986.

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_2))/((x_2 - x_0)(x_2 - x_1)(x_2 - x_3)) xx y_2 + ((x - x_0)(x  x_1)(x - x_2))/((x_3 - x_0)(x_3 - x_1)(x_3 - x_2)) xx y_3` 

y = `((1986 - 1978)(1986 - 1982)(196 - 1990))/((1974 - 1978)(1974 - 1982)(1974 - 1990)) xx 25`

`((1986 - 1974)(1986 - 1982)(1986 - 1990))/((1978 - 1974)(1978 - 1982)(1978 - 1990)) xx 60 +`

`((1986 - 1974)(1986  1982)(1986 - 1990))/((1982 - 1974)(1982 - 1982)(1982 - 1990)) xx 80 +`

`((1986 - 1974)(1986 - 1978)(1986 - 1982))/((1990 - 1974)(1990 - 1978)(1990 - 1982)) xx 170 +`

= `(8 xx 4 xx (-4))/((-4) xx (-8) xx (-16)) xx 25 + (12 xx 4 xx (-4))/(4 xx (-4) xx (-12)) xx 60 + (12 xx 8 xx (-4))/(8 xx 4 xx -8) xx 80 + (12 xx 8 xx 4)/(16 xx 12 xx 8) xx 170`

= `25/4 + (-60) + (12 xx 80)/8 + 170/4`

= `25/4 + 170/8 - 60 + 120`

= 6.25 + 42.5 + 60

= 108.75

∴ Output in 1986 is 108.75 (thousand tones)