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प्रश्न
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 |
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उत्तर
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)
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